LIBRARY Of CONGRESS. 



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f.p3l 



UNITED STATES OF AMERICA. 




LOF 




FOR THE LAKES. 



By H, C. PEARSONS. 



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f" AD 



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CLEVELAND : 

BISSELL &. SCRiVENS^ 

1891. 






Copyrighted 1891 

BY 
BiSSELL & SCRIVENS. 



PREFACE. 



It is probably not apprehended bv more than a very few, 
that in the recent adoption of '^ Standard Time," we have the 
greatest aid to navigation, on the coast and lakes, that could by 
any means have been devised. 

The experienced seaman knows that ^^ direction" is the ulti- 
mate object of all the labored astronomical observations by 
navigators at sea, and that a knowledge of ''place" is precedent 
to that of 'direction." 

The determination of place is the office of nautical astronomy, 
which question alone not infrequently draws largely upon the 
highest powers of the trained mathematician and astronomer. 

But, fortunately, we are relieved of nautical work on the lakes. 
^'The U. S. Lake Survey" has determined the co-ordinates of 
place for all the light-houses on the whole chain of lakes, and 
every ship-master on those waters is furnished with the results of 
this survey, in the ''List of Lights of the Northern Lakes 
and Rivers," for the asking. And as the '* standard time," now 
available in nearly every town in the United States, to the 
fraction of a minute, relieves us of the "chronometer," as used 
at sea, we have only our "lead" and "log" and "compass," to 
look after, but which, as we will find, will give us something to 
do, for the rapid growth in the magnitude of our vessels, result- 
ing from the enormous growth in the volume of our commerce, 
make it imperative that those who are to have the care and 
navigation of them, keep pace in technical and scientific attain- 
ment with those who design and build them. 



IV. PREFACE. 

Accordingly, to help in this matter, I have prepared the 
following Manual of Xavigation for the Lakes* 

Among those features that may be considered novel, are the 
articles on the Dumb Compass, with its application to the finding 
of compass errors; also a method of deducing a table of Time 
Azimuths of the Sun, from an amplitude when the latitude and 
the declination are of the same name. This table is available for 
from one to three hours, requiring but ordinary arithmetical 
knowledge on the part of the ship-master. 

With the admirable charts in use, and with a precise knowledge 
of the place of every light, the finding of compass errors is 
the one problem that suffices for the navigation of the 
lakes. 

The custom of compensating deviation by means of magnets, 
is becoming general, the idea being that a needle brought to its 
place, will remain there indefinitely, and that when it is right on 
the cardinal points, it is also right on all other points. But this 
is a sorry mistake. All compasses, are in error, in some parts of 
the card, even after the most careful adjustment. 

The faces that give position to the needle are so many, and so 
mutable, that it is not safe to depend on the adjustment for more 
than a short time, particularly in a new vessel. 

The remedy lies in adjusting the ship-master. 

To attain a confidence in his compass, a living confidence, born 
of certainty, to meet the growing responsibility laid upon him, 
the ship-master must be educated up to a full understanding of 
this question. 

This subject is not, as too often thought, beyond the reach of 
very moderate attainments, any more than is book-keeping, 
telegraphy, or photography. The diligent application that 
masters the one, will master the other. 

Accordingly, I have dwelt on this topic at some length, giving 
a number of methods of finding compass errors, including that 
by time azimuths of the sun, which, though costing perhaps a 
little more time to master, will in the end be found the most 
efficient and satisfactory of all the methods in use. . 



PREFACE. V. 

It is believed that among the several methods given, of finding 
compass errors, any ship-master of fair capacity will find some 
method that he can readily master, so as to utilize it. 

The article on the ''correction," or compensation of ship's 
compasses, is based on the method taught by such scientists as 
Prof. G. B. Airy, Astronomer Royal; Frederick J. Evans, and 
Archibald Smith, of the Liverpool Compass Committee. 

While I cannot claim this little work to be free from fault or 
imperfection, I yet have a confidence that it will be found of 
sufficient merit to claim respectful consideration from those inter- 
ested in the maritime affairs of the lakes, and to be particularly 
helpful to the young student of navigation, in which last event, 
my highest ambition will have been gratified. 

H. C. PEAESONS. 
Ferrysburg, Mich., Oct. 27, 1891 



CONTENTS. 



PAGE. 

Chapter I. 
Trigonometry 1 

Chapter II. 
The Mariner's Compass 22 

Chapter III. 
The Sailings 40 

Chapter IV. 
Construction of Charts 64 

Chapter Y. 
Terrestrial Magnetism, and the Magnetism of Iron in 

Vessels, etc 82 

Chapter VI. 
The Propeller AVheel 96 

Explanation of the Tables 106 

Table I. 
Traverse Table 112 

Table II A. 
Natural Sines and Cosines 113 

Table II B. 
Natural Tangents and Cotangents 116 

Table II C. 
Natural Secants and Cosecants 119 



Vlll. CONTENTS. 

Table III. 
Trigonometrical and Conversion Table 122 

Table IY. 
The Sun's Amplitude 124 

Table \, 
Sun's Declination, with corresponding Equation of Time. . . . 126 

Table YI. 
Azimuth and Hour Angle for Latitude and Declination .... 128 

Table YII. 
For Eeducing Standard to Mean Local Time 142 

Table YIII. 
Table of Chords to Eadius Unity, for Protracting 144 

Table IX. 
Meridional Parts 146 

Table X. 
For Eeducing Departure to Difference of Longitude 153 

Table XI. 
For the Correction of Middle Latitude, in Middle Latitude 

Sailing 154 

Table XII. 
Distance of Objects by Two Bearings 155 

Table XIII. 
For Eeducing Longitude to Time 156 

Plate I. 
Pearson's Diagram for the Eepresentation of Deviation and 

the Conversion of Compass Errors 158 

Plate II. 
Illustration of the Components of Magnetic Force 160 

Plate III. 
Isoo^onic Chart 162 



A MANUAL OF NAYIGATION FOR THE LAKES. 



CHAPTER I. 

|Trigonometry. 

Signs and Symbols. — The student not already acquainted with 
algebraic notation, should make himself familiar with the follow- 
ing Signs and Symbols, which are among the many used in 
mathematical language: 

= This is the sign of equality; it implies tJiat the quantities 
between which it is used are equal, as 12 inches=:l foot, 3 miles 
=^1 league, etc. 

+ This is called the plus sign, implying an increase or addition, 
as 7+3=10. 

- This is called the minus sign, implying diminution or sub- 
traction, as 7 — S^=4:, 

The plus and minus signs are also used to imply relative direc- 
tion, as to the right or left, up or down, heat or cold, forward or 
backward, etc. 

X This is the sign of multiplication, signifying that the 
quantities between which it is written are to be multiplied together, 
as 4Xo=20. 

-T- This is the sign of diyision, implying that the number prced- 
ing it is to be divided by that one foUow^ing, as 9-^3^=3. 

i ) ] h [ ] These are called brackets. Their use is to indicate 
that the quantities embraced within them are to be considered as 
one quantity, thus, 5 (5+3)— 6 (3+2)==10. 

This is another form of bracket, and called a yinculum 

or bar, and is used when two or more quantities already connected 



A MANUAL OF NAVIGATION FOR THE LAKES. 



by brackets, are to be regarded as one quantity, as 5 (5-|-3)— 6 
(5^^)-^5 (8— 6)--2=5. 

: :: : These are the signs of proportion. They are a modifica- 
tion of the signs of division ^-, and equality ^, and may be 
written thus, -^ = h-. They signify that the first term divided 
by the second=the third term divided by the fourth. Quantities 
having this relation are called proportional quantities or numbers, 
as 4: 8:: 3: 6,, or better, 4-T-8=3-r-6=J. The value in this case, J, 
is called the ratio of the proportion. 

A power of a number is indicated by a small figure above and 
to the right of the number, thus, 42=16 indicates that the number 
4 is used twice as a factor. 4^^64. 

A root is indicated by the symbol ;/, which alone implies the 
second or square root. . A small figure written over the symbol, 
thus, ,/, implies the degree of the root, thus, ,/64=4. 

Symbols of CJuantity. — To abreviate mathematical investiga- 
tions, quantities or numbers are represented by some symbol, as a 
letter, — usually the initials of the quantity; and the symbols of 
operation already explained, aj^ply to them the same as to the 
numerical quantities. 

Lines are separated by letters at different points in their extent, 
usually at their extremities. Thus, the line between the points 
A B is called the line AB. 

Angles are defined by giving one letter in each of the lines 
containing tlie angle, together with the letter at their intersection, 
thus, the angle between the two lines ^"^^^1^^==^^ AB and BC, is 
written ABC, the middle letter indicating the angular point. 

Numerals are represented by single letters, the leading letters of 
the alphabet representing known quantities and the final letters 
. unknown quantities. 

The use of an equation is, generally, to bring out the value of 
some unknown quantity, which is the object of inquiry, and 
which is involved in the equation. Thus, in a proportion, one of 
the terms is usually an unknown or required quantity represented 
by one of the terminal letters (usually x) of the alphabet, as 

A:B::C:X. 
From this, by the principle well known in arithmetic, that the 
product of the extremes equals the product of the means, we have, 

AX=:BC, 
in which X is the quantity desired. 



TRIGONOMETRY. 6 

Then, frOm the well known axiom, that wc may multiply or 
divide equal quantities by the same number without destroying 
the equality between those quantities, we may divide both sides 
of this equation by the factor, when we have the equation, 

X=BC^A, 
in which the quantity represented by X is equal to the product of 
the two quantities B and C divided by the quantity or factor A. 
This operation is called solving the equation for X, and the 
expression, X=BC^-A, is called a formula. 

Alg'ebraic Addition is the aggregation into one sum, of quan- 
tities affected by the opposite signs of -}-j — ? or plus and minus. 
Thus, if a man has ^100 in bank to his credit, or -{-, and he has 
bills payable, — , ^90, his estate is only $10 — that is to say, the 
sum of Dr. and Cr. sides of his % is + f'lOj the two signs can- 
celling each other to the extent of the smaller number. 

Algebraic Difference is found by changing the sign of the 
quantity to be subtracted, then combining the quantities as in 
addition. Example: The difference of latitude between 10° X. 
and 5° S., or +10° and — 5°=_15°. By changing either sign, 
both terms become ^^like," i. e.j both -f- or both — , and they 
come together by addition, so that we may have +15° or — 15° as 
the difference of latitude of the two places. 

Trigonometry is that branch of geometry that treats of the 
relations of the sides and the angles of triangles, both plane and 
spherical. In this work, only plane triangles will be considered. 

In all the wide range of mathematical science, there is, perhaps, 
no one branch so generally useful or of so wide an application in 
general affairs of life, as trigonometry. 

To the navigator, the astronomer, surveyor, geographer and 
the civil engineer, it is simply indispensible. Xatural philosophy, 
mechanics, optics and geology could not be treated without this 
branch of mathematics. 

It is hoped the little I may introduce of the subject, and which 
is as little as we can do with, will induce the student not already 
familiar with it, to pursue it further in some of the many excel- 
lent standard works on that science. 

Whatever we may give of trigonometry, we shall treat without 
the use of logarithms, — the object being to make the subject as 
elementary as possible. 



4 A MANUAL OF NAVIGATION FOR THE LAKES. 

There are two kinds of magnitude considered in triangles, viz., 
au^les and sides or lines. 

For the purpose of measuring angles, a circle is introduced. 
The circumference of this circle is supposed to be divided into 
360 equal parts, called degrees; these are further divided into 60 
equal parts, called minutes; and these, again, into 60 equal parts 
called seconds. 

The degrees are indicated by a small circle ° over and to the 
right of the figure; minutes by a dash ^, and seconds by two 
dashes ^^ above and to the right of the number of minutes or seconds. 
Thus, 15 degrees, 28 minutes and 46 seconds would be written, 
15°, 28^, W, etc. 

A degree or a minute, then, is not a magnitude of length, but 
merely a certain part of the whole circumference, without regard 
to lineal dimensions; it is merely ratio. 

In measuring an angle, the center of the measuring circle is 
located at the intersection of the two lines that contain the angle 
to be measured. The arc intercepted by the two lines is the 
measure of the angle. 

But, as degrees, etc., and lines are of different kinds of mag- 
nitude, they cannot be directly compared. Moreover, the mag- 
nitude of an angle does not vary with the size of the triangle. So, 
to make these elements comparable, auxilliary lilies have been 
introduced in and about the arc to be measured, or considered, in 
such a manner as to make them depend not only on the magnitude 
of the arc, but on that of the triangle to be considered. 

These auxilliary lines are called fuiictioiis of the arc, and have 
received different names according to the different positions or 
relations they have to the arc. 

Names of the Functions. These are eight in number, as 

follows (see Fig, 1): 

In the triangle ADC, let the arc AB be the measure of the angle 
ACD. Then (1) the Sine is the perpendicular BG drawn from 
one extremity, B, of the arc, to a diameter or radius drawn 
through the other extremity of the arc. 

(2). The Tangent, AD, is drawn perpendicular to the diameter 
through one end of the arc, to meet a line drawn from the center 
of the circle through the other end of the arc, as at D. 

(3). The Secant, CD, is a right line from the center through 
one end of the arc, to meet the tangent, as in D. 



TRIGONOMETRY. 



(4). The Cosine, CG, is that part of the radius intercepted 
between the foot of the sine and the angular point C. 

(5). Tlie Tersine, AO, is that part of the radius intercepted 
between the foot of the sine and the extremity of the arc, or the 
foot of the tangent. 

The Compliment of an arc or angle is what the angle lacks of 
being a full quadrant, or 90°. Thus, the angle ACE, being a 
right angle or 90°, the arc BE is the compliment of the arc AB. 
The prefix ^'co'^ being used to show that the function before 
which it is used pertains to the compliment of the arc under 
consideration. Thus: 

(6). The Co-Tangent, 
EF, is the tangent of the 
compliment of the arc AB. 

(7). The Co-Secant, 
CF, is the secant of the 
compliment of the arc AB. 

(8). The Co-Yersed 
Sine, E H, is the versed 
sine of the arc AB. 

The Supplement of an [ 
arc or angle, is what the 
arc lacks of 180°. Thus, 
the arc BCI, embracing 
the angle DCI, is the sup- 
plement of the arc AB. 
And it has the same sine 
BG as the arc AB. 

The Sine and Cosine 
connect the two arcs AB and BE, so that when one is known 
the other is also known, for it will be observed that the sine of 
one of the arcs is the cosine of the other. The student will observe 
that the sine and tangent of an angle are always opposite the 
angle. 

Relations Between the Parts of a Triangle. — It is shown in 
geometry that the sum of the two acute angles of a right plane 
triangle is equal to two right angles, so that when one of the 
two is known the other is also known. 

The Radius is any right line drawn from the center to the cir- 
cumference of the circle. The radius diminished by cosine is 




Fig. 1. 



6 A MANUAL OF NAVIGATION FOR THE LAKES. 

yersiue (see AG in Fig. 1), and diminished by sine, is CO- 
versine, EII. 

In tables of trigonometrical functions, radius being unity, all 
the internal functions, as sine, cosine, etc., are decimal fractions, 
as also the tangents of arcs less than 45°, and the co-tangents of 
arcs greater than 45°. This is more particularly shown in the 
following table of the 

Limits of Trigonometrical Functions : — Let AB be any arc 

(see Fig. 1) varying from at B to 90° at E; then when 

AB=^0, the sine=0, the versine^O and cosine=:l, 

^^ tan.=0, " secant=l and cotan.=infinity, 
AB=90°, the sine^l, the versine=l and cosine=0. 

the tan.=infinity, the secant=infinity and cotan.=:0. 
AB=45°, the tan.=l and cotan.:=L 
AB=60°, " cosine=J and versine=^. 

The student should make himself familiar with this table, so 
that he can tell the limits of any function on hearing it called. 
He should also observe the effect of any change in the arc AB. 
Thus, if AB be increased, the radius remaining the same, then 
all the primitive or direct functions will be increased, while the 
co-functions will be diminished. Again, if the radius AC be 
increased, the angle or arc BA remaining the same, then all the 
functions will be increased in the same proportion. 

Thus it is seen that no change can be made in the magnitude of 
the angle or in the radius of the measuring circle, without a cor- 
responding change in the value of all these auxilliary lines. So it 
is the introduction of these lines that has brought the solution of 
triangles within the pale of arithmetic. 

The two sides containing the right angle are called the legs. 
Sometimes base and perpendicular. But any side may be made 
the radius, i. e., the side by which the others are measured, when 
the other sides take names according to their relation to the angles 
of the triangle: Thus, in the following triangle, ABC, if BC is 
made the unit for measuring the other sides, then AB is sine and 
AC is the cosine of angle C. 

Again, if in Fig. 3, AC be made the radius, or measuring unit, 
then AB becomes the tangent and BC the secant of angle C. 

Or if, as in Fig. 4, the perpendicular AB be made radius, then 
the base AC becomes tangent and BC the secant of the angle B. 

The student should make himself familiar with these changes. 



TKIGONOMETRY. 




Fig. 2. 




Fis:. 3. 



The relations existing between the functions of arcs, are those 
resulting from the comparison of similar triangles, and consist 
of equality of ratios, which is a fundamental principle of trig- 
onometry. 

Thus, in Fig. 5, the triangles ABC 
and A^B^CS having their sides 
opposite the equal angles A, A ^ ; B, 
B^, etc., parallel, are similar. 

Also the triangles ABC and A^ 
B^E (Fig. 6), and ADE of same 
Fig., having their "like" sideS 
parallel, are all similar triangles. 
" Like Sides '' are those that are 
opposite equal angles. They are 
also called homologous sides; thus, 
BC and DE, Fig. 6, are homologous 
sides, being opposite the angle A, 
with the sides BC and DE parallel. 
Again, the sides AD, A^B^, being 
parallel, and opposite the same angle 
E, are homologous. 

The ratios of these homologous 
sides is the same between any two 
*'like'* sides of similar triangles, i. e., 
triangles be divided or measured by a 
triangle, the quotient will be the same 
as that of any other side in the same 
triangle, measured by its corresponding 
''like'' in the other. 

This relation for a right triangle is 
expressed thus, for any angle, as ACB 

BC AC AB 

DE AE AD 
Cosine Sine R 




Fi-. 4. 



if any side of one of the 
'like" side in the other 

Fig. 5. 



R Tan. Secant C 

These relations are the foundation of 

the several rules for the solution of 

right triangles, and w^hen stated in the 




Fig. 6. 



8 



A MANUAL OF NAVIGATION FOIl THP: LAKES. 




form of a proposition, are as follows : 

Cos.: R:: Sin.: Tan. I Sin.: Cos.:: H: Cotan. 
::K:Sec. j ::R:Cosec. 

Here, it will be seen, are five quantities besides the right angle, 
— three sides and two angles — any three of which being given, — 
one being a side, — the other two may be found. Hence, 

In a right angled triangle, — the two acute angles being compli- 
ments of each other, — there are only four parts to be con- 
sidered, viz., three sides and one angle. 

Some two of the six ratios of the preceding article will always 
contain the two given quantities and the required quantity, And 
we have the four following cases, viz.: 

(I). Given, the Hypothenuse and An- 
gles, to find the two sides containing the 
right angle. 

(II). Given, the Hypothenuse and one 
Side, to find the two angles and the other 
side. 
(III). Given, the two Angles and one 
Fig. 7. Side, to find the hypothenuse and the 

other side. 

(lY). Given, the two Sides, to find the angles and the hypoth- 
enuse. 

To Solve a Right Triangle, for any of its parts, it is only 
necessary to select any two of the six ratios that contain 
the two given elements or parts and the required part of the 
problem, and arrange them in the form of a proportion, then 
Solve the proportion precisely as in arithmetic, by multiplying 
together the extremes and the means of the proportion in two 
products, and divide both products by the factor that is found 

connected with the required 
quantity. 

Example. Suppose we wish to 
find the hypothenuse of a right 
triangle, the angles and one side, 
— say the perpendicular, — being 
given. 

Solution. Let ABC be the tri- 
^^' ' angle whose side b is required, — 

the angles and side AB being given. 




TKIGONOMETE, Y . 



It will be found convenient to represent the angles hy the cap- 
ital letters and the sides by the small letters of the same name as 
the opposite angle. 

Making either side, say a, the radius, the given side c is the 
tangent of the angle C. Also, the required side is the secant of 
angle C. Thus, our two given parts are R and tan., and the 
required part is see. of angle C. 

Then, as in arithmetic, form a proportion, so that the ratio of 
the functions will form the first couj^let, and that of given and 
required side the last couplet, thus: 

Tan. C;Sec. C::c:b. 
Multiplying extremes and means, as in arithmetic, 

Tan. Cb=Sec. Cc. 
Dividing both products by first term, as in arithmetic, 

Sec. C c 

b= (1). 

Tan. C. 

Again, making b the radius, then, 

Sin. C : K : : c : b, and, as before. 
Sin. C b=:Rc, then dividing and K being unity, 
C 

b=-— (2). 

Sin. C. 
Again, making AB or c the radius, 

R : Cosec. C : : c : b, or 
Eb=Cosec. Cc. 
AVhence, by dividing, and remem- 
bering that R is unity, wc have, 

b=Cosec. Cc /o\ 

=3Sec. A. ^''''* 

From the above, it will be seen 
that we may make any side of the 
triangle the measuring side, but to 
make the given side, as c, or the re- 
quired side, as b, in the above ex- Fig. 9. 
ample, the measuring unit will give the simple solutions, as seen 
in equations (2) and (3), above. 

Thus, to find the side BC or a, we would in the same manner 
have, 

a==c tan. A, 




10 A MANUAT^ OF NAVIGATION FOR THE LAKES. 

or, if we make a the radius, then, 

c 
Si= ■ 



tan. c. 

In this manner many formulas have been constructed for the 
solution of right jDlane triangles. From them I have selected a 
few of the most useful, and which are sufficient for the solution 
of such questions as are likely to come before the navigator in 
*' dead reckoning." 

Preparatory to the solution of practical questions, the student 
must be informed of some of the Trigonometrical Tables and 
their uses. 

Table of the Natural Functions. — These are the Sines, Co- 
sines, Co-tangents, the Secants and Co-secants, Yersines and Co- 
versines of arcs varying by one minute, for a whole quadrant. 

They are called natural functions to distinguish them from the 
same functions when given by their logarithms. 

This table gives all the above elements to radius unity, — thus, 
we have all the three sides and their corresponding angles for 
5400 right triangles; and by considering the angles to vary by 
parts of a minute, as by 10^^ or 20^^ we could readily deduce all 
the elements for many thousand more triangles. 

So, having the elements of any proportioned triangle to radius 
unity, that can come before us in practice, we have only to insti- 
tute a proportion between the known parts of one triangle and 
the corresponding parts of the tabular triangle, and reduce the 
proportion to find the parts desired. 

These sines, tangents, secants, etc., are arranged in columns, 
under their rbspective names. 

The degrees for arcs of less than 45° are found at the top of the 
page, with the minutes in a column at the left. For arcs greater 
than 45°, the degrees are found at the foot of the page, with the 
minutes in a column at the right. (See table II). 

Let the student look out the following functions. (Our table 
gives them only to intervals of 5°) : 

The natural sine of 25°, 40^= .4331 
" '' cosine '' " = .9013 

'' " tangent '' '' = .4805 

" '' cotangent '' '' =2.0809 

'' '' secant '' '' =1.1095 



TRIGONOMETRY. 11 

The natural cosecant af 25°,40^=2.30S7 
'' '' sine of 74°, 25^= .9632 

" " cosine '' '' == .2686 

'« " tangent " " =3.5856 

" '' cotangent '' '' = .2789 

'« " secant *' '^ =3.7224 

'' '' cosecant '' " =1.0382 

The equations spoken of in a former article, as being selected 
for this work, are the following: 

Formulas for the Angles of a Right Triangle. 

-, J Sin. C=c^-b^==perpen(licular-i-hypothenuse. 

' \ Cos. C::=:a^-b^base-^hypothenuse. 
o r Tan. C^=:c-7-a=perpendicular-T-base. 

*\Cotan. C=a-4-c=zrbase .-perpendicular. 
J Sec. C=bH-a=hypothenuse-T-base. 

* I Cosec. C=b-i-c=hypothenuse-i-perpendicular. 

It will be observed that the preceding equations are in pairs, 
€ach pair finding an angle by a function with its co-function. 

It will be observed, too, that each angle is found indirectly by 
i3ome function, as sine, cosine, etc., so that the angle correspoding 
lo the function must be found from the table of sines, cosines, etc. 

It will be observed, also, that any function is found by dividing 
one side by some other side of the triangle, which is for the pur- 
pose of reducing the sides of the triangle in question, to radius 
unity, for the purpose of making them comparable with a tabular 
triangle. 

It will also be observed, in the second member of each of the 
equations in the preceding article, that some one side of the tri- 
4ingle is divided by some one of the other sides. Then, turning 
this divisor over to the other member of the equation, as a 
multiplier, we find a side of the triangle. In this manner we 
derive the following: 

Formulas for the Sides of Right Triangles. 

r at=b cosine C=hypothenuseXcos. C, or sin. A. 

• \ =c cotan. C=perpendicularXcotan. C, or tan. A. 
rb=ra sec. C=baseXsec. C, or cosec. A. 

• \ =0 cosec. C=perpendicularXcosec. C, or sec. A. 

r c=b sin. C=hypothenuseXsin. C, or cosine A. 

* \ =a tan. C==baseXtan. C, or cotan. A. 

(See Fig. 10). 



12 



A MANUAL OF NAVKxATION FOR THE LAKES, 



Geometry gives the following equations for the sides 
triangle, viz.: 

-(1). 

-(2). 
-(3). 



)f a 



a^:r^b- 
b 



r|/rad.- — sin. 2 C 



y^a--t-c2=r|/sin.- C+C08.- C- 




c^=/b2 — a2=-|/rad.2— oos.2 C 

The student should make himself so thoroughly familiar with 
the three sets of equations in each of the three preceding articles, 
^^ that he can readily select any one 
wanted for the occasion. 

With regard to numerical work, the 
student, having made himself familiar 
with the preceding articles, is prepared 
for the solution of plane riglU triangles, 
but preparatory to a numerical solution, 
he should be able to solve them by coii- 
Jd struction, which involves the use of a 

Tal)le of Chords of Arcs, or a 
^^S' 10« Protractor. — A scale of chords is 

sometimes engraved on the ordinary drafting scales. But it is 
to a particular radius given on the scale, — and therefore of but 
limited use, — moreover, it is not sufficiently precise for good 
work, though a good protractor w^ill do. 

By a principle of geometry, the chord of an arc is twice the 
sine of half the arc. Whence, a table of chords can be con- 
structed directly from a table of natural sines. Thus, 

Eequired to find the chord of 26°, 28^. The sine of the half of 

this arc, or 13°, 14^, is .22892. Twice this decimal, retaining four 

places of figures, is .4578, which is the chord of 26°, 28"^ to radius 

unity. (Our tables, varying by 10^, will give the chord for 26°, 

'30^=.4584). 

In this manner was the table of chords computed. It is arranged 
with the degrees at the head of the columns and the minutes in 
columns at the margin of the page. And it is computed for 90°, 
varying by 10^ (See table YIII). 

Thus, the chord of 43°, 10^= .7357 
'' '' 60°, 00^=1.0000 

'' '' 84°, 40^=1.3469 

'' " 4°, 05^= .0713 

" " 15^= .0044 



TRIGONOMETRY. 



13 



In using a table of chords for the construction or the measure- 
ment of angles, the student should have a scale divided decim- 
ally. Then, by removing the decimal point one or two places to 
the right, he may have a large scale to set off his decimals with, 
and thus attain great precision. 

He will also want compasses for ink and pencil, a T square and 
triangle or parallel rule. If not already provided with such instru- 
ments, he should send twenty or thirty cents to some dealer in 
drawing instruments, for an Illustrated Catalogue. 

Solution of Plane Triangles.— Examples : 

1. Sailed S. 48° W., 126 miles. 
How far did I sail south, and 
how far west ? 

By Construction. Draw a 
vertical line AB (see Fig. 11) to 
represent the meridian from 
which the course is reckoned, A 
representing the starting point. 

Look in the table for the chord 
of 48°, which we find to be .8135 
to radius unity. Removing the 
decimal point one place to the 
right, we have 8.135 as the chord 
of 48° to radius 10. 

Take 10 units to any convenient scale and set them off from A 
to m. Then take the chord 8.135 to the same scale and set them 
off from m to n. 

Through the points A and n, draw an indefinite line, and on it 
set off the distance AC, 126 miles, to any convenient scale. 

Through C, draw a line at right angles to the line AB, meeting 
that line in B. 

Measure the lines AB and BC by the same scale as that by 
which AC was set off, and we have the measure desired of the 
two sides. 

AB or c=southings=84.31 miles. 
BC or a==westings==93.63 miles. 

By Computation. In this problem, it is seen that we have 
the angles and hypothenuse from which to find the sides a and c, 
containing the right angle. 




Fig. 11. 



14 



A MANUAL OF NAVIGATION FOR THE LAKES. 



By reference to the first equation of the third pair of equations, 
(page 9), we see that the perpendicular c is found by multiplying 
the hypothenuse b into the cosine of the angle A, adjacent to c, 
or into the sine of C, the angle opposite, or 
c=bXcos. (A=48°) 
=126 X .6691=84.31 miles southings. 
In the same manner from the first equation of the first pair, we 

have, 

a=bXsine (A=48°) 

=126 X .7431=93.63 miles westings. 
It will be seen that the numerical work is only indicated here. 
The student should look the functions from the tables and per- 
form the multiplication; and he may also construct the angles 
with a protractor, though the method by ^'chords" should be 
learned. 

2. Example: Sailed due north 146 miles, then due east 76 
miles. Required the course and distance back to the place sailed 
from. 

By Construction. Draw vertical line on the page to represent 
the meridian, and set off on it, to 
any convenient scale, the distance 
sailed north, 146 miles, C to B. 

Then turn off a line at right an- 
gles to the right, and set off on it 
the distance sailed east, 76 miles, 
B to A, then join A with C. 

AC, measured on the same scale, 

will be the distance required. We 

are now to find the angles at A and 

C. We could apply a protractor 

and read the angle directly from it, 

but it is better to measure it by 

means of its chord to some radius — 

Fig. 12. say 100. Thus, 

Take 100 in the compass, to any convenient scale, and sweep an 

arc from the side a to the side b, with C as the center. Also from 

the side b to the side c produced, from A as a center, sweep the 

arc m, n, for the purpose of measuring the angle A. 

The object of measuring or computing both angles is to have a 
check on our work. The student will soon see abundant reason 




TRIGONOMETRY. 15 

for this in the non-pr.ecision of his first efforts. A and C together 
should=90°. 

The chord measuring the angle C to radius 100, will be found 
=47.5, and that for A=103.7. 

Kemoving the decimal point, in each case, two places to the 
left, for the purpose of finding the chord for radius unity, we 
have for C, .405, and for A, 1.098. Taking these chords to the 
table for the angles, we find A=62°, 30^ and C=27°, 30^ and A 
C=164.6. 

As courses are always measured from the meridian, we must 
take the compliment of the angle A for our return course, which 
is S. 27°, 30" W. and distance=164.6 miles. 

By Computation. In this case we have the two sides contain- 
ing the right angle to find the angles and the hypothenuse. By 
equation 2 (page 9), we have. 

Tan. C^^perpendicular-r-base, 

=76^146=.5205=tan. (27°, 30^=C). 
Thus, dividing the perpendicular by the base, we get the tangent 
of the angle opposite the perpendicular, — in this case, the decimal 
.5205. Then, looking in the table of natural sines, tangents, etc., 
we find this decimal in column of tangents, under 27° at the head, 
and in line of 30^ on the left. 

One of the two angles being found, we may subtract it from 90° 
for the other-=62°, 30^ 

But, as a check, it is better to compute it in the same manner, 
by the same rule, thus: 

146-^-76=31. 92105=tan. (A-=62°, 300- 

From equation (2) of (page 9), we have, 
b=a X secant (C-=27°, 30^ 
=146 X 1.1274=164.6=distance. 
Examples: 

3. Given the hypothenuse 26. The angle at the base=26°, 
20''. Bequired base and perpendicular. (See eq. 3 of page 9). 
Ans. Base=23.37. 
Perp.=11.38. 

Note. — Let the student construct graphically, with care. He 
should use a sharp lead pencil, making a line as small as can be 
clearly seen, and using a needle point for pricking off distances. 



16 A MANUAL OF NAVIGATION FOR THE LAKES. 

4. Given the hypothenuse 146. Angle at the base, 72°, 30''. 
Find perpendicular and base. 

Ans. Base=43.95. 
Perp.=89.28. 

5. Given hypothenuse 84, base 46. Required the perpendic- 
ular and the angles. (See pages 8-9 for angles). 

Ans. Angle at base==o6°, 50'' 

^' '' vertex=33°, 10^ 
Perpendicular=70.29. 

6. Given hypothenuse 218, one of the sides contain the right 
angle=46. Required the angles and other side. 

Ans. Angle opposite the greater leg=77°, 50^ 
'' '' '' smaller '' =12°, 10^ 

The other leg=213.13. 

7. One leg or side=76, angle opposite=43°, 28^. Required 
the hypothenuse and other side. 

Ans. Hypothenuse=110.5. 
Other Side, =80.18. 

8. One side=243, adjacent angle=18°, 40^. Required the 
hypothenuse and other side of the triangle. 

Ans. Hypothenuse=256.49. 
Other Side, = 82.08. 

9. Given hypothenuse 180.3, legs 176, and 39.04. Required 
the angles of the triangle. See group of equations for angles, 
(page 8). 

Ans. Angles, 12°, 37^ and 77°, 23^ 
Let the student find an angle, say the smaller one, from each of 
the three sides, thus: 

Hypoth.H-perp.=cot. angle at base=12°, 37^. 
Perp.H-base=tan. angle at base=:12°, 37^. 
Hypoth.^-base=sec. angle at base=12°, 37^. 
Note. — As our tables vary by 5^, the nearest minute given by 
them for this case will be 12°, 35^, and as our compasses are not 
graduated to appreciate angles smaller than 1°, the practice of 
computing, using 5 or 6 place decimals with intervals of V and 
finding all angles to the nearest V , is simply a waste of time, — 
accordingly, we have abridged the tables, and thereby the work. 
Oblique Plane Triang-les. — The solution of oblique plane 
triangles will require the use of a few formulas different from 
those required for right plane triangles. 



TRIGONOMETRY . 



17 




Fig. 13. 



Proposition 1. In any plane triangle, the sides are propor- 
tioned to the sines of the opposite angles. 

In the oblique triangle ABC, if we make the two sides AC and 
BC radius, the perpendicular CD is the sine of each of the angles 
A and B. Then, 

AC : CD : : K : sin. A, or AC sin. A=E CD. 
And BC : CD : : K : sin. B, or BC sin. B=E CD. 
But the second members in each 
of the above equations are the same, 
whence the other two are propor- 
tioned, and we have, 

AC sin. B=BC sin. B, or 
AC : BC : : sin. A : sin. B, 
according to proposition. 

Solution by the Properties of Ri^ht Triangles,— three 
€ases. — As all cases of right triangles may be solved by the prop- 
erties of right triangles, we give no more rules for their direct 
solution, though a number are known. 

I. The three parts given may all be adjacent, as when two sides 
with their included angle are given. 

In this case, demit a perpendicular from the extremity of the 
smaller side onto the other side, as from C to D in the triangle 
ABC. 

In the right triangle A C D, we 
have hypothenuse and one side, with 
the angles from which to find the 
perpendicular CD, which may be 
found by equations on page 9, as 
also may the base A D. 

In the triangle BCD, subtracting ^^^' -*^^* 

AD from AB, we have BD. Then, CD having been computed, 
we have the two sides containing the right angle from which to 
find the angles. 

The angles A and B, being now known, we have only to take 
their sum from 180°. when we have the angle at C, on the principle 
that the sum of the three angles of any plane triangle is 180°. 

The side BC remains to be found. We may find it by Prop. 1, 
just explained, and called the sine proportion, or we may find it 




18 A MANUAL OF NAVIGATION FOR THE LAKES. 

by means of the equations for the sides of aright triangle, page 9» 
By the ''sine proportion," the solution is, 

Sin. B : AC : : Sin. A : BC. (Page 12). 

II. When the Angles and One Side are Giren. — From one 
extremity of the given side, demit a perpendicular onto one of the 
other sides, or onto one of them produced, as from C to D on the 
side AB produced, of the triangle ABC. (Mark with a dash, — , 
the given parts). 

In the right triangle ACD, we have all the angles and the 
hypothenuse AC given, from which we may find AD and DC^ 
from rules already explained. 

Then, in the right triangle DBC, the angle at B is known,, 
because it is the supplement of B in the 
oblique triangle ABC, whence both angles 
of the triangle BDC are known, and DC 
having been computed, BC can be found 
from equations, page 9. 
,_. Now, AB may be found from the ''sine 

proportion," Prop. 1, page 12, when the 
triangle is determined. 
The proportion for BC is. 
Sin. B : AC : : Sin. A : BC, 
and for AB it is. 

Sin. B : AC : : Sin. C : AB. 
Thus the case may be solved either by the rules for right triangles 
or by the sine proportion. 

III. When the Three Sides are given, to find the Angles* 

The solution of this will depend on the following: 

Proposition 2. In any plane triangle, as the greater side 
: The sum of the other two sides, 
: : The difference of the other two sides, 
: The difference of the segments of the greater 
side made by a perpendicular from the 
opposite angle. 
In the triangle ABC, let AB be the greater side. With AC as 
a radius, and from C as a center, sweep the arc EAF and produce 
BC to E. Then BE is the sum of the two smaller sides, and BF 
is their difference. 

AB is the sum of the two segments of the greater side, made by 
the perpendicular CD, and BG is their difference. 




TRIGONOMETRY. 



19 



Then, because the two lines BA and BE are two secants, cutting 
the circle and meeting outside the circle, the whole secants and 
their external segments are inversely proportional, and w^e have 
the proportion, 

AB : BE : : BF : BG, 
which is the proposition. 

Then, half of this difference added to half the base, or greater 
side=tlie greater segment. And half this difference sub- 
tracted from half the base, gives the smaller segment. 

This being done, the triangle 
is divided into two right triangles, 
in each of which the base and 
hypothenuse are known, from 
which the angles are determined 
by the equations on pages 8-9. 

Example: In the oblique tri- 
angle ABC, given the three sides, 
64, 56, 34, to find the angles. 

Solution: Demit a perpendic- 
ular from C onto the greater side ^ ^S- ^^* 
at D, dividing that into the two segments m and n, then by the 
last proposition we have, 

As AB (=m+n)=64 
:AC hCB=90 
::AC— CB=22 
: m— n=::90X22^64=30.9416. 

Then, J (m— n)=15.4708, 

1 (m+n)=32.0000. 

Whence, m =47.4708, 

and n =16.5292. 

The oblique triangle is now divided 
into two right triangles, in each of 
Avhich the base and hypothenuse are 
given to find the angles, which can be done by means of the equa- 
tions on page 8. 





Then, the sum of the angles A and B being taken from 180° 
before explained, we have angle C. 



as 



20 



A MANUAL OF NAVIGATION FOR THE LAKES. 



Examples of the three cases of oblique triangles for solution. 
1. Given the two sides, 91 and 60, with their included angle, 
42°, 30^. Required the other side and the other two angles. 

Solution: First, by Construction. In the triangle ABC, lay 
off one side, preferably the longer side AB, to any convenient 
scale=to the given side, 91. From each point, A and B, sweep 
an arc on a radius of 10 or 100 units, — say 100, to any convenient 
scale, for the purpose of measuring the angles at B and C. 

On the arc opposite A, set off the chord of 42°, 30^, from n to 
c, and draw the line A c. On this line, set off from A, the dis- 
tance AC:=60, and join 
CwithB. Then is the 
triangle constructed. 

Measure BC by the 
same scale, and that side 
is determined=(61.8). 

Produce BC to b, and 
measure the chord mb=^ 
70, w^hich corresponds 
to (40°, 560 for the an- 
gle at B. Then 180° — 
(BTC=383°, 260-^(96° 
34^)=angle at C. 

Verify by construct- 
ing C thus: With the 
same radius as for the 
other angles, sweep an arc from C as center and produce, if neces- 
sary, the CB and CA to it, as at o and p. We find the chord o p 
greater than the chord of 90, the limit of the table of chords, so 
we measure the arc in two parts, and find it to be (96°, 34^), as 
before, or verify with a protractor. 

By Computation. In the triangle 
ACD, Ave have the angles and one 
side (the hypothenuse) to find the 
other sides. 

Then multiplying AC by sine of 

A, we have DC or .6756X60=.(40.54 

Fig. 19. =DC). Eq. 3, page 9. And ACX 

cos. 42°, 300=AD; or .7373X60=(44.24=AD). Eq. 1, page 9. 

And AB-AD=91— 44.54=(46.76=BD). 





U^.2^ 



^6.7^ 



TRIGONOMETRY. 21 

The angle at B is found by its tangent, by dividing DC by DB, 
or 40..j4^46.76=(.S670=tan. 40°, oij'). Eq. 2, page 8. 

The side BC is found by multiplying the side or base BD by 
sec. of angle at B, or 46.76X1.324=(61.85=BC). Eq. 2, page 9. 

The angle at C is found by taking the supplement of the sum of 
the angles A and B, which gives (96°, 34^=C). 

The student should complete the multiplications indicated in 
the above computations, looking the functions out of the tables, 
and he should make careful constructions to scale, of all his prob- 
lems of triangles, as the geometrical figure will always help to a 
clear conception of the numerical work required. 

There are many other theorems for the solution of oblique 
triangles, but those given are deemed sufficient for all of that class 
of questions likely to occur. 

Examples of Oblique Triangles. — 

1. Given one side b of a triangle=32 rods. The angle A 56°, 
20^, and the angle at C=49°, 10^ to find the sides a and b. 

Ans. a=27.5 rods, 
b=25.0 " 

2. Given the angle A=63°, 35^, the side b=32, and the side a 
=36. Required the other side and the other two angles. 

Ans. Angle B=o2°,45J% 
" C=63°, 30.}^, 
Side AB=36, nearly, 

3. Given angle A=26°, 14% the side b=78, the side c=106. 
Eequired the angles B and C, and the side a. 

Ans. B=43°, 14% C=110°, 2% side a=oO. 

4. Given the side a^o4, c=78 and b=70. Required the 
angles. 

Ans. A=42°, 22% B=60°, 52t% C=76°, 45V. 



CH.APTER II. 



THE MAHINEK S COMPASS. 



Concerning the early history of this instrument, we have but 
Jittle reliable information. 

We first hear of it in China, where the needle or 'Moadstone'' 
was used to give direction, by floating it on a piece of cork, on 
water. 

Then we hear of it sus2:>ended at the middle by a string, when 
the magnetic force of the earth would give it direction. 

Early in the 14th century, the compass was improved by placing 
the needle on a '^pivot post;" and in 1608, the Rev. Wm. Barlow 
further improved it by applying the ^'gimbal joint" to the bowl 
carrying the needle. But we must consider it as at present con- 
structed. 

The Needle is made of steel, hardened and magnetized. It is 
then surmounted by a card graduated into 32 points, and some- 
times into quadrants of 90° each, and sometimes into degrees, con- 
tinuously, from at the north, around by the east, south, west, 
to 360° at the north. 

The better class of compasses have two to four needles under 
the card. 

Names of the Points. — The line passing through the point on 
which the needle swings, and the north point of the card, is called 
the Zero, or Meridian Line, — the intersection of this line with 
the north part of the card being marked with a ^ 'fleur-de-lis," by 
way of distinction; the south side being marked by the letter S. 

The line at right angles to this meridian, is called the '^ Equa- 
torial," from its analogy to the equator, — its extremities being 
marked with the initials of the directions they represent, — E for 
east, and W. for west. 

The four letters, N., S., E., W., are called the cardinal letters, 
and the points they represent are called the four Cardinal Points., 
Sometimes the Equatorial is called the Prime Yertical. 



THE COMPASS. 23 

SubdiTiding" the Card. — The points midway between the car- 
dinal points are called the Inter-Cardinal Points, sometimes 
the Quadrantal Points. 

A quadrantal point takes the name of the two cardinal points 
between which it lies. Thus, the point midway between the north 
and east, is called the N. E. point, and so of the other quadrants. 

In the same manner, the point midway betw^een the cardinal 
point and a quadrantal point, is indicated by calling the letters 
between which it lies, — as N., N. E., indicates the point midway 
between the north and the northeast, and so for the other octants. 

The remaining sixteen points are found adjacent to the cardinal 
letters and the quadrantal letters, one on each side of each letter, 
the adjacency being indicated by the word *^ by." 

In defining one of those points, the reference letter is first 
named, then the point immediately to the right or left of the 
reference letter, as the case may be. Thus, the first point imme- 
diately to the right of east, is called E. by S.; that to the left of 
north, N. by W., etc. Thus, the plan of naming the points being 
understood, the name of any point of the compass is easily 
brought to mind. 

The naming of the points of the compass in consecutive order 
around the card, is called "Boxing the Compass," and the stu- 
dent should make himself so familiar wath it that he can name 
them readily in either direction. 

Reading the Compass Card by Numerals. — In giving the 
course on which the ship is to be steered, the officer of the deck 
usually refers the same to the nearest cardinal or quadrantal letter. 
But in making computations for bearings, or for reduction of 
course, or the finding of compass errors, reference is always made 
to the nearest meridian letter, N. or S., and the bearing is reck- 
vpned in points and parts of a point, or in degrees, toward the east 
or west, as the case may be. 

This is made necessary by the fact that all trigonometrical tables 
used in calculating courses, are arranged in that manner. 

The student must therefore make himself familiar with reading 
the card in that manner. Thus, 

E. by N, would be read, N. 1 p. E. 

N. E. '' '^ N. 4 p. E. 

E. JN. " '^ ^r. 7f p. E.,etc. 



24 A MANUAL OF NAVIGATION FOR THE LAKES. 

The student should also be able to change a course given by its 
numerical value to its cardinal name. Thus, 

N. 3 p. E. would be called X, E. by X\ 

s. 6 p. ^y. '^ '' w. s. \v. 

N. 5p. W. '' '' X. W, byW. 

X. 45° E. '' '' X. E. 

S. 33f° E. '' '' S. E. by S., etc. 

Table III will facilitate this reduction of courses, and it is 
important that the student make himself expert in these reduc- 
tions. Some compass cards are graduated both in degrees, and in 
points and \ points. Such a card is then, in itself, a table for 
the reduction of course from one denomination to the other. 

The Use of the Compass, is to give the bearing of a line, or 
to point out the direction in which a ship is to be steered. All 
courses are ultimately referred to the astronomical meridian, when 
they are called True or Astronomical Courses, sometimes they 
are called Chart Courses. 

But the compass seldom points to the true north. In most 
places it is turned away from that direction by the earth's mag- 
netism, more or less. 

It is also disturbed from the position which it assumes under 
the earth's magnetism, by the magnetism of the iron in the ship. 
This is the most serious disturbance that can come to the compass 
needle, — for as the ship heads to different courses, the needle also 
takes different directions, — so that the change in the readings of 
the card is no indication of the change made in the course of the 
ship. Yet, at sea, in dark weather, the navigator is obliged to 
refer his course to the disturbed needle, — whence it is of the first 
importance that the navigator know the error of his compass. 

The Ag'OUic Liue, or line of no variation, is a line on which 
the magnetic needle, when not disturbed by local causes, points 
to the true north. 

This line, in the United States, commencing in South Carolina, 
takes a course a little to the west of north, crossing the head of 
Lake Erie near the mouth of the Detroit Kiver, thence through 
the eastern part of the State of Michigan, crossing Lake Huron a 
little west of Mackinac Island, thence across the eastern part of 
Lake Superior to a little east of the Copper Islands. 



THE COMPASS. 25 

From this chart it will be seen that east of the agonic line, the 
north end of the needle stands to the left, or west of the astron- 
omical north; and to the west of this line, the north end of the 
needle points to the east of the true meridian. Thus, in the Gulf 
of St. Lawrence, the needle points 20° to the left of north, while 
on the Pacific coast, near the mouth of Columbia Kiver, the north 
end of the needle stands to the right of the true north, 22°. 
Thus, in crossing the continent, the north end of the needle 
swings toward the east nearly four points. 

Preparatory to finding the error of the needle, Ave give a few 
definitions: 

Astronomical Meridian. — This is the north and south line 
given on all charts, and to which all courses are eventually 
referred. The meridian of any place is in the plane containing 
the earth's axis, and that place. 

Mag'netic Meridian. — This is the line pointed out by the mag- 
netic needle, when acted upon by the earth's magnetism alone. 

Compass Meridian. — This is ;he line pointed out by the mag- 
netic needle, acting under the combined influence of the earth's 
magnetism and that of the iron in the ship. 

Variation, is the difference between the directions of the 
astronomical meridian and the magnetic meridian. The north end 
of the needle may stand either to the right (east) or to the left 
(west) of the true meridian, in which case the variation is, 
called east or Avest, as the case may be. Easterly variation is con- 
sidered + (plus) and westerly — (minus), and is usually given in 
degrees. 

Deviation, is the difference between the directions pointed 
out by the magnetic and the compass meridians. The compass, 
when aboard ship, may be disturbed by the ship's magnetism, so 
as to take position to the right or left of the magnetic meridian, 
precisely as the magnetic meridian takes position to the right or 
left of the true meridian. Whence, 

Deviation is east or west, + or — ? precisely as in variation 

Variation and deviation are thus seen to be strictly analogous,' 
but with this difference, viz.: Variation is constant for all head- 
ings of ship, but deviation is different for all headings of ship. 

Three Kinds of Bearing. — Bearing being the direction of an 
object or place, with regard to some line of reference, as a mer- 



26 A MANUAL OF NAVIGATION FOR THE LAKES. 

idian line, it follows from the above definition thai we have three 
kinds of bearing, viz.: 

Astronomical, or True Bearing*, which is the direction of an 
object, with regard to the true meridian. 

Magnetic Bearing is the direction of an object or place, with 
regard to the magnetic meridian; and 

Compass Bearing" is the direction of an object or place, as 
given with the deviated needle. Whence, we have at sea: 

Variation and Deviation combined into one error called Cor- 
rection or Total Variation, and which is the algebraic sum of 
variation and deviation, for all compasses at sea are under the 
influence of both, — the earth's magnetism and that of the ship. 

That part of compass error due to variation, is found at the 
time of making a survey of the coast, or of the lake, or of any 
locality, and recorded in the charts of the same; and is regarded, 
for the time, as constant, though it changes slowly by the slow 
change in the ' 'agonic" line. Thus, in 1840, Prof. Elias Loomis 
informed us, in the American Journal of Science, that the agonic 
line, commencing at a point near Wilmington, N. C, went north- 
westerly, crossing Lake Erie east of Cleveland; thence, through 
the middle of Lake Huron and entirely east of Lake Superior. 
The variation in 1890, or rather the agonic line, was from 1J° to 
2° to the east of what it was in 1840, i. e., places that then had 
variation, have variation of 1J° to 2° west, now. As a conse- 
quence, the variation that was given on charts that were made 
from early surveys, are in error, according to the age of their 
surveys — say 1° to 1J° — westerly variation increasing and easterly 
variation decreasing. 

But not so with that part of compass error due to deviation. 
This must be found for the compass of each individual ship, and 
for compasses in different parts of the same ship. Nor can we tell 
from the behavior of a compass in one ship, what may be looked 
for in the compass of another ship. 

Azimutli, is the angle or course by which an object is referred 
to a meridian. In geodesy, it is usually reckoned from the south 
part of the meridian (in north latitude), and from to the right, 
by the west to the north, and east 360° back to the south. In 
navigation and surveying, it is reckoned from both parts of the 
meridian, 90° each way to the east and west. 



THE COMPASS. 27 

Amplitude, is the bearing of an object when referred to the 
east or west point of the compass. 

Azimuth and Amplitude are compliments each to the other, i. 
€., each is what the other lacks of eight points, or 90°. 

Swinging Ship for Compass Errors. — That part of the com- 
pass error due to deviation, is found by "swinging ship,'' and 
<jomparing the reading of the ship's compass with the simultaneous 
readings of a "shore compass" on successive headings, as the ship 
turns around in azimuth. 

The "shore compass" is the ordinary compass used by surveyors, 
but sent on shore to get it away from the influence of the ship's 
magnetism to Avhere it is acted on only by the earth's magnetism. 
The line between the ship's compass and the "shore" compass is 
a reference line. The bearing of it, for successive headings of 
ship, compared with the bearing given by the "shore" compass, 
gives the error (deviation) of the ship's compass on the several 
headings. 

Standard Compass. — The above method of finding the error 
of ship's compass, involves the use of some appliance on board the 
ship for measuring the angle between the ship's heading and the 
Teference line. 

At sea, this appliance is found in the movable ring and movable 
sights attached to the standard compass, which is placed on the 
open deck, and high enough so that bearings can be taken over 
the rail, even when the ship is heeled over a few degrees. 

But on our lake vessels there are no such appliances. The com- 
passes are not only not provided with the ring and sights, but 
they are boxed up in the pilot house where such things could not 
he of any avail if we had them. As a consequence, our lake ves- 
sels, as a class, are utterly without the means of finding their 
compass error. 

The Dumb Compass has been prepared to meet this deficiency. 
It is the movable ring and sights of the standard compass, as used 
at sea, but in another form. It is not necessarily expensive, — a 
good one may be made by the ship's carpenter, — though better by 
the instrument maker. 

The essential feature of the dumb compass is that it may be 
read by two indexes. It has no needle. The card is graduated 
like that of the ship's compass. 



28 A MANUAL OF NAVIGATION FOR THE I.AKES. 

The lower index corresponds precisely to the ^^lubber's" mark 
of the ordinary compass bowl, — the line joining it with the center 
of the compass to be fixed parallel with the center line of ship, as 
is that of ship's compass. 

The plate on which the card is fixed is movable to any position 
about a vertical spindle, and may be clamped in any position. 

The sights are on a bar that turns in azimuth about the same 
spindle as that carrying the card, and which can be clamped to 
the card in any position. 

It should be mounted on a tripod and be provided with a socket 
joint, and a level vial should be fixed in the plate carrying the 
card, for adjusting the card to a horozontal position; and it should 
be set up high enough so that a clear view can be had from it of 
the whole horizon. 

(XoTE. — The ideal instrument would be on ^^ gimbals,' ' but 
this is not necessary, or even important, for, as the instrument is 
not used for a steering compass, but merely as an auxilliary com- 
pass, it is quite sufiicient if it be mounted on a tripod provided 
with a ball and socket joint, and with cross levels in the card). 

The Use of the Dumb Compass, is to find the angle between 
the heading of the ship and any line that is used as a reference 
line in finding compass errors. Thus, in port, w^hen it is desired 
to find compass errors by reference to a line of known magnetic 
bearing, the process is as follows: 

1. Send the shore compas-s ashore and set it up at any con- 
venient point where it is away from any local disturbance of the 
needle, and take the bearing to the ''dumb compass" on board 
ship. 

2. The dumb compass being set up on deck in a suitable place 
for observation, the zero line and lubber line being parallel with 
ship's center line, and '' looking forward," the same as the ship's 
compass. 

3. Set the upper index, or sights, to the same reading as that 
given by the shore compass, the north side of the card being to 
the north. 

4. The card and sights being clamped together, turn them 
together till the sights " back. sight " on the shore compass. Then 
is the card of the dumb compass oriented to the magnetic mer- 
idian. The reading of it by the lower index is the magnetic head- 



THE COMPASS. 29 

ing of the ship. And the difference between the magnetic head- 
ing and '^compass" heading is the deviation, if any, of the ship's 
compass, for that heading. This is called the method by Recip- 
rocal BeariugS, and is used in harbors or other places where a 
distant azimuth mark cannot be found. 

Finding Compass Errors. — The illustration of the use of the 
dumb compass in the preceding article, gives the method of finding 
compass errors in port, but the office of that compass is the same 
for all methods of finding compass errors, viz., to refer ship's 
head to the reference line, — which may be any line of known 
bearing, — the results only being different, as when the difference 
line is known by its magnetic bearing or by its astronomical 
bearing. 

Direct Method, hj Reference to a Distant Object. — In this 
case, the work of finding compass errors is somewhat simplified. 
The azimuth mark may be a distant building, headland or shore 
line, but it must be so distant that the change in the place of the 
compass, in swinging ship, will not cause any appreciable parallax, 
— say two or three miles, or not less than about 100 times the 
diameter of the circle made by the compass in swinging ship^ 

The bearing of this line is found as in the preceding article, or 
it may be found from a chart, but when found, the subsequent 
work is that of the former case, except there are no reciprocal 
bearings to make with the shore compass, — whence, it is called 
the direct method. 

Method of Recording Obseryations of Compass Errors and 
deriving therefrom the Deviation. — The method of recording 
the observations for finding deviation of our compasses, is some- 
what different from that at sea, for the reason that our compasses 
are differently mounted and differently equipped from those at sea. 

There, the bearing of an azimuth mark or reference line can be 
directly compared with the bearing of the same line, as indicated 
by the ship's compass, and on the same instrument. But our com- 
passes, as before seen, are so arranged that the bearings of objects 
cannot be taken with them, they can only show the apparent bear- 
ing of ship's head, hence the necessity of a different form of record. 

Example: The table on the following page, gives the observa- 
tions of the Steamer Huron, swung for deviation, at South Haven, 
by the author, June 26, 1876. (See form I). The first column 
gives the bearings of ship's head, as indicated by her compass. 



30 



A MANUAL OF NAVIGATION FOR THE LAKES. 



The second column gives the bearings of ship's head, as indicated 
by the ^ 'shore compass" (or rather by solar transit with index 
correcting for variation). The third column gives the difference 
of these two columns, which is the deviation for the respective 
bearings or headings of ship. 

It will be seen by referring to columns 1 and 2, form I, that the 
first course in the second quadrant of the first column, corresponds 
to the last course of the first quadrant in the second column. Now, 
in finding their difference, we must reckon both courses from the 
same zero point, — no matter which — say the north, in this case. 
Then, instead of S. 6 p. E., we would have X. 10 p. E., i. e., v/e 
must take the supplement of the course. Then we can find the 
difference of the two bearings, 2h p. 





I. 




II. 








Deviations of 


D 


EVIATIONS OF 






Steamer Huron. 


Steamer Huron. 






Head by 
Ship's 


Head by 

^^Shore" 


Devia- 
tions. 


Head by 
Ship's 


Head bv 
^SShore'' 


Devia- 
tion o 




Compass. 


Compass. 


Compass. 


Compass. 


LiLi 


' 




Xorth. 


N. fp.E. 


fp.E. 





f 


~Tv 


. E. 


1 


X.lip.E. 


N.H E. 


i W. 


H 


li 


i 


W. 


X.2| E. 


N.ll E. 


i w. 


2f 


H 


* 


W. 




N.4i E. 


X.2f E. 


n w. 


4i 


2| 


u 


W. 




N.5 E. 


X.3} E, 


If w. 


5 


3} 


If 


W. 




N.6 E. 


X.3I- E. 


2J w. 


6 


3J 


2* 


W. 




N.7| E. 


^M E. 


2f W. 


7i 


oj 


2f 


W. 




S.6 E. 
S. 5 E. 


N.7J E. 


21 \y. 

2i W. 


10 
11 


8f 


2J 

2i 


W- 




S.7J E. 


W. 


2 


S. 3f E. 


S.5J E. 


2f W. 


12i 


lOJ 


2i 


w. 




S.3 E. 


S.4X E. 


H w. 


13 


11f 


li 


w. 




S.2 E. 


S.3J E. 


IJ w. 


14 


121 


IJ 


w. 




S.l E. 


S.2 E. 


i w. 


15 
161 


14 

16 


1 


w. 




S. i W. 


South. 


i w. 


w. 




S. i W. 
S.lf w. 


S. f w. 
s.lf w. 



J E. 


16f 


16f 









I'f 


l"f 


i 


E. 


3 


s.2 J W. 


s.3 J W. 


1 E. 


18J 


m 


f 


E. 




S. 3 W. 


S.3i W. 


1 E. 


19 


191- 


|- 


E. 




S.3t W. 


S.4| W. 


U E. 


19| 


20f 


1f 


E. 




S.41 W. 


S.5J W. 


If E. 


201 


211- 


If 


E. 




S.5^ W. 

S.6J ^y. 


S.6J W. 


If E. 
If E. 


21J 
22J 


221 
241 


1^ 
If 


E. 




N.7f W. 


E. 




S.7f w. 


N.6| W. 
N.4| W. 


H E. 
2J E. 


23f 

25J 


25f 
271 


1* 

28 


E. 




N,6- W. 


E. 


4 


N.3i W. 


N.IJ W. 


If E. 


28f 


301 


If 


E. 




N.lf W. 


N. ^ W. 


1* E. 


30f 


31J 


u 


E. 



THE COMPASS. 31 

And so in the third and fourth quadrants. 

Writing the Courses by Azimuth. — This method is given in 
form II of the preceding table, which consists in writing them 
continuously in points, from zero at the north, around by the east, 
etc., to 32 points. 

This method is very simple, when the card is graduated con- 
tinuously around the circle, and greatly facilitates the finding of 
the difference of the courses given by the two compasses. On that 
account, it is desirable to reduce the courses, as ordinarily given, 
to azimuth readings, as follows: 

1. In the first quadrant, i. e., from the north to the east, 
write the courses as they are given from the card, — the courses 
being given by their numerals. 

2. In the second quadrant, E. to S., write the supplement of 
the compass reading. 

3. In the third quadrant, S. to W., add 16 p. to the compass 
reading. 

4. Add 24 p. to the compliment of the compass reading, in 
the fourth quadrant. 

Thus, the azimuth reading for S. 2 p. E. is 14 p.; for S. 2 p. 
W., it is 18 p.; for N. 7 p. W., it is 25 p. 

We remind the student that supplement is what the course 
lacks of 16 p. or 180°, and compliment is what it lacks of 8 p. or 
90°. 

Naming the Deviations, will require a little careful attention, 
from the fact that each two adjacent quadrants are read in opposite 
directions; which may cause the young student to stumble in this 
matter, if he is not on his guard. 

He should first go over the table and write the difference of the 
courses given in the first and second columns, without regard to 
name or sign in the third column. Then consider which way 
(right or left) the card of the ''shore" compass must be turned to 
make its readings agree with those of the ship's compass. That 
direction is the name to be applied to the differences for the 
deviations. 

Thus, in the first quadrant, the readings of the ''shore" com- 
pass for any given course, are less than those of the ship's compass 
for the same course, and if we turn the card of the "shore" compass 
to the leftjthe two readings may be made to agree, — i. e., the devi- 
ation is west by the amount of the difference of the readings. 



32 A MANUAL OF NAVIGATION FOR THE LAKES. 

Again, in the second quadrant, the reading of the *^shore^' 
compass for any particular course, is greater than that of the 
ship's compass, and yet the card of the * 'shore" compass must be 
turned to the left to make the readings of the two compasses 
agree, — so that there, also, we have westerly deviation. 

The same results will be found between the third and fourth 
quadrants. This comes from reading the alternate quadrants in 
opposite directions. 

Deviation Curve. — We are told by the books, to take the devi- 
ations on the consecutive points, when swinging ship. But this, 
while possible, is at times totally impracticable, and we take such 
as we can get. Thus, it will be seen that the headings given of 
the Huron, in the preceding table, are very irregular as to their 
intervals. Still, it is important to have the deviation for regular 
intervals, as by points, so that the deviation for intermediate 
courses, as for parts of a point, may be interpolated for steering 
purposes. 

Many schemes have been devised for the attainment of this 
object, — and mostly of the graphical type. Among the best of 
these methods, are those of J. E,. Napier, Esq., F. R. S., and 
Archibald Smith, M. A., of the Liverpool Compass Committee. 
(See Admiralty Manual, for 1874). 

Some sixteen years ago (1891 now), the author of this Manual 
devised a scheme for the representation of deviation and the con- 
version of compass courses, which on being submitted to Prof. 
J. H. C. Coffin and Commodore A. W. McCormick, U. S. X., is 
endorsed and recommended by both of these eminent men, as 
being a decided improvement on that of Mr. Napier, accordingly 
I give the method of constructing it and of using it. 

Pearson's Diagram, was invented for the representation of 
deviation, and for the conversion of compass errors. 

This diagram (see plate I) shows two scales, — one vertical and 
one horizontal. 

The vertical scale represents azimuth, and is to be read down- 
wards, — such reading corresponding to the reading of the compass 
card from the left to the right, as we read a watch dial. It is 
simply the circumference of the card, developed by rolling it down 
the page and marking the points. 



THE COMPASS. 33 

This scale may be read in points and parts, or in degrees, as 
desired, — the horozontal lines across the page being the gradua- 
tion marks. 

The horizontal scale, — or rather the two scales, one at the 
head, the other at the foot of the page, — is for setting off varia- 
tion and deviation. 

The thick line drawn vertically through the middle of the page, 
represents the magnetic meridian, to which deviation is referred. 

The thin lines drawn obliquely across the page, from the left, 
upward and to the right, are for the purpose of performing addi- 
tion and subtraction graphically, of variation and deviation. 

A thin line drawn vertically through the page, parallel with 
the magnetic meridian, and cutting the variation scale at any 
point, is called the local meridian, corresponding to that 
variation. 

Any two convenient scales may be taken, at pleasure, for the 
variation and the azimuth scales, — the condition being that the 
oblique lines for performing addition and subtraction, pass 
through points in each scale, having the same numerical value. 
Thus, if an oblique line pass through the azimuth line at, say N. 
1 p. E., then it must also pass through the variation or horizontal 
scale at 11^° from the zero of that scale, etc. 

It is observed that easterly variation is placed to the left of the 
magnetic meridian, and westerly variation to the right. This is 
for the purpose of locating the magnetic meridian to the right of 
the true meridian for easterly variation, and to the left of the true 
meridian for westerly variation, as it should be. 

But, deviation being referred to the magnetic meridian, must 
be set off from that line, according to its name, — to the right for 
easterly and to the left for westerly deviation. 

Construction of the Deviation Curve. — The deviations having 
been satisfactorily found,- as on page 29, set them off to the right 
or left of the magnetic meridian, according to their name (see plate 
II). Thus, when ship's compass needle indicates north, it is known 
to be out of place to the right by f p. Take f p. in the dividers 
from the variation scale at head of page, and set it off to the right 
on the line of north, from the magnetic meridian, and make a dot 
surrounded with a small circle, thus ©. 

Again, when ship's head is IJ p. E. of N., the needle is out J p. 
to the left. As before, take J p. in the dividers and set it off 



34 A MANUAL OF NAVIGATION FOR THE LAKES. 

on azimuth, IST. IJ- p. E. from the magnetic meridian, making the 
mark (©) as before. 

In this manner, set off all the deviations, according to their 
name, — to the right for easterly deviations and to the left for 
westerly deviations, on the horizontal lines corresponding to their 
azimuth, and from the magnetic meridian as zero point. 

Through the points set off, draw a fair freehand curve, giving 
and taking a little at times to make the curve fair, and you have 
the desired deviation curve. 

It will be observed that about half of the curve is on either side 
of the magnetic meridian. On this account, this deviation is 
called semi-circular. 

It will be observed, too, that the two parts are about equal, as 
the curve crosses the meridian when ship heads N. by E., also 
when it heads S. by W., but this is not always the case. And when 
they are not equal, it is not always that good reversals can be had 
in compensating the compass for deviation, as will be seen here- 
after. 

Use of the Deviation Curve, Reduction of Courses. — The 

deviation curve being constructed, we are prepared to solve many 
questions that vex the ship-master who is compelled to work with 
a deviated compass. A few examples will illustrate: 

Example 1. At Grand Haven, Mich., mean var. 3° E. It is 
desired to sail to Milwaukee, with Steamer Huron. Eequired 
steering course for ship's compass, also the return steering course. 

Solution: First, draw a line through var. 3° E., to represent 
the true or chart meridian. On this line, take up the chart course 
required to be made, viz., W. J S., as at a, aud project it onto the 
curve at b, at W. by S. I S., which is the compass course desired^ 
— Ans. S. 6J p. W. 

To find the return course: Reverse the given chart course, W. 
J S., to E. J N., and take it up on the true meridian as before, as 
at c, and project the same onto the curve at d, when will be found 
the required compass course for the return, viz., E. S. E. J S., or 
S. 5 Jp. E. 

Example 2. On the same diagram will be seen the deviation 
curve of the U, vS. S. Monadnoc. Required the outward and the 
return steering courses for the compass of this vessel for the same 
voyage. 



THE COMPASS. 35 

Solution: As with Example 1, take up the chart course, W. i 
S., on the true meridian, as at a, and project the same onto the 
curve of the Monadnoc, as seen at b^, in azimuth W. J N.^N. 7J 
p. AV. 

For the return course. Reverse the chart course, W. J S., to 
E. J X., and as before, project the same onto the curve, as at c^, 
where we find the required return course, viz., E. by X. f X., or 
N. 6i p. E. 

Discussion of the Preceding Problem. — It will be remem- 
bered that w^hen the card is turned to the right of its normal 
place, the readings for the bearing of a given object are too small, 
when regarded as azimuth (as all courses are eventually regarded), 
and consequently we must take up a course smaller than the 
compass would indicate by the amount of disturbance, which 
deduction is made by moving up the page of the diagram; other- 
wise it would take us too far to right. 

For the same reason, when the card is out of place to left, its 
readings for the bearing of an object are tOO large, and without 
reduction, the card would take us too far to left. So that in this 
case we must increase the readings by the card by the amount of 
disturbance, to attain any given course. This is attained by mov- 
ing down the page. 

Tariation, is seen to be the distance between the magnetic 
meridian and the true meridian. 

Deviation, is the distance between the magnetic meridian and 
the curve, at any point measured on the horizontal line passing 
through the point, and by the variation scale. 

" Correction," or Total Error, is the distance between the 
true meridian and the curve. 

These quantities always come together algebraically, westerly 
deviation and variation being called minus ( — ), and easterly are 
called plus ( + ). 

Correction for Leeway, can also be made by means of this 
diagram. 

Example 3. At Grand Haven, Mich., mean var. 3° E. Required 
the outward and the return compass courses for each of the 
steamers Huron and Monadnoc, taking into consideration a port- 
hand leeway of | p. — the chart course to Chicago being S. W. J S. 

Solution: Take up chart course, S. W. J S., on the true mer- 
idian (3° to left of magnetic meridian) and move down the page 



36 



A MANUAL OF NAVIGATION FOR THE LAKES. 



I p. to compensate leeway. Then project the course thus found, 
viz., S. W. i W., onto the curve at e, when we find S. W. by S. 
for the Huron; and projecting onto the curve of the Monadnoc, 
we find S. W. J W. for that vessel's compass. 

Keturning, with the same wind, our leeway will be to starboard, 
then moving up the page f p. to N. E. f N. (the course having 
been reversed) and going to the curves, we find for the Huron, 
N. E. f E., and for the Monadnoc, N. N. E. J E., for their res- 
pective compass courses for the return voyage. 

Example 4. At Duluth; var. 8° E. ; wish to make a chart course 
N. E. by E. J E., and looking for a port-hand leeway of J p. 
What is the compass course of the Huron and for the Monadnoc, 
as deduced from their respective deviations? 

Solution: On a true meridian drawn through var. 8° E., take 
up the desired chart course and move down the page J p., to 
compensate the port leeway. Project the point thus found, onto 
the curve for each vessel, and we find for the Huron, E. J X. at 
g; and for the Monadnoc, N. E. f E. at g^, as their respective 
steering courses. 

Deviations of Steamer Huron. 



Head by 




Head by 




Ship's 


Deviation. 


Ship's 


Deviation. 


Compass. 




Compass. 




North. 


7FE. 


South. 


5 ° W. 


N. by E. 





s. by ^y. 





N. N. E. 


6 W. 


s. s. w. 


5J E. 


N. E. by N. 


11} W. 


S. W. by S. 


11 E. 


N. E. 


17 W. 


S. W. 


13i E. 


N. E. by E. 


21 W. 


S.W.bvW. 


14 E. 


E. N. E. 


24 W. 


W. S. W. 


18} E. 


E. by N. 


26* W. 


W. by S. 


21 E. 


East. 


28^ AV. 


West. 


23 E. 


E. by S. 


28-1 ^y. 


W. bv X. 


24J E. 


E. S. E. 


28" W. 


W. N. W. 


24J E. 


S. E. by E. 


27 W. 


X.W.byW. 


24 E. 


S. E. 


25 W. 


X. W. 


22 E. 


S. E. bv S. 


21J W. 


X.W.byN. 


19^ E. 


s. s. e: 


18 W. 


X. X. W. 


15 E. 


S. by E. 


12 W. 


X. bv W. 


lU E. 


South. 


5 W. 


Xorth. 


6" E. 



It is believed the above illustrations will fully explain the oper- 
ation of reducing compass courses. 



THE COMPASS. 37 

Another use of the curve, besides the reduction of courses, is 
found in the facility with which it gives the Deviation on Con- 
secntive Points of the card when observations have been made 
at irregular intervals, thus making the deviations available for a 
steering card. To do this, we have only to measure the deviations 
on each point, by taking the distance between the curve and the 
magnetic meridian, with a pair of dividers, refer them to the vari- 
ation scale at the head of the page, and tabulate the results. 
Thus, we find at north, the deviation curve to the right of the 
magnetic meridian. Applying the distance to the scale at the 
head of the diagram, we find it 7J°, which we write in the devia- 
tion column, with its name, E. 

Shaping the Course. — The four questions just discussed con- 
stitute the problem of ''shaping the course," and assumes the 
following form: With a compass that is deviated, what course 
shall be taken to attain a given chart course ? 

Rule, using the diagram: Project the given chart course from 
the true meridian onto the curve. At the intersection, is found 
the compass course desired, to attain the required chart course. 

To Find the Course ''made good," is to find what chart 
course has been actually attained, by sailing with a compass that 
is deviated. 

Knle. Project the compass course from the curve onto the 
true meridian. At the intersection will be found the equivalent 
chart course. 

This problem, which is seen to be the reverse of the first ques- 
tion, is wanted at sea, in making up the ''day's work" for finding 
place of ship, but is seldom or never wanted on the lakes, for the 
reason that the place of ship, when outside, is rarely, if ever, 
looked for, as it is looked for at sea. 

The runs being short, — only a few hours at most — points, as 
lighthouses, are seen so frequently, and their co-ordinates of place 
being known from ihe list of lights, there is no necessity for 
"reducing a traverse," as at sea, to find place of ship, and no 
necessity for finding the "course made good." 

The Log-Line and Time-Glass. — These are also essential parts 
of the equipment for keeping account of the place of ship at sea. 

The log-line is adapted to the sea-mile of 6080 feet, with a time- 
glass of 28 seconds. 



38 A MANUAI. OF NAVIGATION FOR THE LAKES. 

Theoretically, the time-glass was supposed to run 30 seconds 
and the sea-mile to be 6087 feet, but as the circle of the equator is 
larger than any other great circle of the earth, vessels always 
found themselves ahead of their reckoning, by using the equa- 
torial mile as the unit of distance. 

To correct this inconvenience, the time-glass has been reduced 
to 28 seconds, and the sea-mile has been reduced to 6080 feet, 
which corresponds now nearly with the minute of a circle on the 
mean diameter of the earth. 

These values are now adopted by nearly all maritime people. 
Then taking the same part of 6080 feet, that 28 seconds are of one 
hour, the length of the ^'knot" on the log-line would be 47 feet. 
Some navigators make it more or less, accordingly as they think 
their ship over-runs or under-runs her reckoning. 

The U. S. navy make the length of the log-line 45 feet, for a 
28 second glass. 

On and about the great lakes, the statute mile of 5280 feet is 
regarded as the unit of measure for distance. Accordingly, for a 
28 second glass, the knot on the log-line should be 40 feet, or 43 
feet for a 30 second glass. 

If the patent log, adapted to the sea-mile be used, its readings 
may be reduced to indicate statute miles, by multiplying the indi- 
cated distance by 1.15, which is adding 15% to the indicated 
distance. 

As the modern patent log depends on the *'pitch^' of its screw 
to give it the right number of turns for indicating the distance, 
the labor of getting the right pitch is somewhat tedious, it being a 
tentative operation. 

But it is not necessary that it should show the number of turns 
for the correct distance. It is quite enough to know the number 
of turns made for a known distance. 

Suppose, for a distance known to be 76 miles, the dial indicates 
80 miles, — that is, it indicates 80-76ths of the distance. Invert 
this fraction and make 76-80ths of 80 miles, for the correct dis- 
tance, i. e., multiply the indicated distance by the reciprocal of 
the screw's rate. 

Thus, suppose the log shows only 13 miles when it should show 
15 miles. Then, inverting the ratio 13-15ths to 15-13ths and 
multiplying by 15, we get the correct distance. Whence, 15-13ths 



THE COMPASS. 39 

or 1.154 is the factor or co-efficient by which to multiply all indi- 
cations of this log. This is called finding the co -efficient of 
the log. 

Distance by Propeller Wheel. — Another method of measur- 
ing the ship's rate of sailing, is that by means of the propeller 
wheel. 

The vessel, when in her usual trim, is run over a known distance, 
noting the total time and the number of revolutions per minute. 

The known distance being reduced to feet and divided by the 
total number of revolutions in the given time, gives the ''net 
pitch" of the wheel in feet, i. e., the distance made good by one 
revolution. 

Example: A steamer making with her wheel 106 revolutions 
per minute, for 64 minutes, makes a known distance of 12.4 miles. 
Bequired net pitch of wheel. 

Solution. 12.4X5280 ^ ^^ , ^. . -, ^ -, , 

—^j-^-rjr~^=z9 .QQ feet. Jset pitch of wheel. 

The net pitch being known, and multiplied by the number of 
turns per minute, and by 60, the number of minutes in an hour, 
then divided by the number of feet in one mile, 5280, gives the 
rate of ship in miles per hour, — or if he divide by 6080 feet, he 
will have the rate in nautical miles per hour. 

In this manner the ship master can readily log his ship for dif- 
ferent speed of engine, and thus learn his rate of sailing to a good 
degree of certainty, and better than by a log whose ''rate" is not 
known. 



CHAPTEK III. 

The Sailings. 

Navigation implies the conducting of a ship from one port to 
another, and in its broadest sense, implies the solution of many 
problems involving a knowledge of geography, nautical astronomy 
and mathematics, to a large extent. 

The terms Plane Sailing, Traverse Sailing, Parallel Sailing, 
Middle Latitude Sailing, Mercator's Sailing, Current Sailing, etc., 
indicate different methods of finding place of ship, rather than 
any peculiarity in the manner of sailing, as the name might seem 
to imply. 

The result of any of these methods is called the place of ship 
by dead reckoning', or the method of finding place of ship from 
the course, the distance, the rate, time sailed, and a knowledge of 
the place sailed from. It is also called the place of ship by account. 

Plane Sailing, is the method of finding place of ship with 
regard to place sailed from, by means of the co-ordinates, course 
and distance, from the properties of the plane triangle. 

The method of plane sailing has been denounced as being inac- 
urate, but when we recollect that the course or ^' rhumb line'' 
makes a constant angle with the several meridians w^hich it crosses, 
we see that distance sailed, the difference of latitude, and the 
departure, are correctly represented by the hypothenuse and sides 
of a right triangle, in which the course is the angle opposite the 
departure, and the hypothenuse is the distance. We give a few 
definitions: 

Equator. — A plane through the center of the earth and at right 
angles to the earth's axis, is called the plane of the equator; and 
the intersection of this plane wdth the surface of the earth, is 
called the equator. 

Zero, or Prime Meridian. — Any plane through the earth, and 
containing the earth's axis, is called a meridian. Any and every 



PLANE SAILING. 41 

point on the surface of the earth has its meridian. The meridian 
passing through the observatory of Greenwich, is now assumed as 
the prime meridian, or meridian from which longitude is reckoned 
by most maritime people. 

Greo^raphic Latitude, is the distance in arc, measured on the 
meridian of any point, from the equator, — or the angle between 
the plane of the equator and a vertical or plumb line passing 
through the point; but, 

Geocentric Latitude, is the angle included between the plane 
of the equator and a line joining any point on the surface of the 
earth with the center of the earth. Geographic latitude is always 
slightly greater than geocentric latitude, in consequence of the 
unequal diameters of the earth, this excess amounting to about 
llj minutes at lat. 45°. Geocentric latitude is wanted in finding 
longitude from the place of the moon, but geographic latitude is 
always used in giving the place of any point, with regard to the 
equator. 

Long'itude, is distance in arc on the equator, east or west. ..Its 
use is to refer any point on the surface of the earth to the prime 
or zero meridian. Latitude and longitude together, are called the 
geographical co-ordinates of place. 

Course, is the angle between the meridian and the track of ship, 
and is usually represented by its initial, C. Distance, is the 
length of line sailed; and course and distance together, are called 
the polar co-ordinates of 2)lace, — the course referring to the mer- 
idian, and the distance referring to the place sailed from. 

But this is special. It concerns only the two points mentioned, 
— the place of ship and the place sailed from, — whence, the method 
by geographical co-ordinates, is general. It not only refers the 
places to each other, but to the equator and the prime meridian, 
and thence to any other point on the surface of the earth. 

Whence, before we can compare the location of places as found 
by ^'course and distance," with that as determined by latitude 
and longitude, we must reduce our polar co-ordinates to geograph- 
ical co-ordinates. This reduction is made by means of a Traverse 
Table, which may be called a table of right triangles. 

In sailing on any oblique course, we make two components, — 
one with regard to distance on the meridian, and one with regard 
to distance east and west, on the prime vertical. 



42 A MANUAL OF NAVIGATION FOR THE LAKES. 

The component, north and south, is called the difference of 
latitude; and that made on the east and west line, is called depart- 
ure. Just what these components are for any course and distance, 
is shown by table I. 

This table is computed for courses varying by J point, and for 
distances varying by unity up to 10. 

By removing the decimal point one place to the right, in any 
column, we multiply the sum by 10; moving two places, we mul- 
tiply by 100, etc. 

By this device, the table is made to give the distance likely to 
be sailed at any one run, and to occupy but a small part of the 
space it otherwise would. 

The courses are given up to four points in the left hand column, 
— increasing from at the head to four points at the foot. 

On the right hand, they are given from four points at the foot, 
to eight points at the head. 

The components are abbreviated at the head of the columns, to 
diff. lat. and dep., for the courses in the left hand column, and at 
the foot of the column for the courses given in the right hand 
column. 

It will be observed that the column that is marked lat. at the 
head, has dep, at the foot. 

The Manner of Using the Table, will be seen from the fol- 
lowing examples: 

1. Sailed N. N. E. J E., or N. 2^ E., 57 mHes. Kequired the 
components of the course, i. e., the northings and the eastings, — 
or diff. lat. and dep. 

Opposite 2 J p. in the left hand column and under 5 in the lat. 
column, we find 4.52. Eemoving the decimal point one place to 
the right, we have 45.2 miles as the northings for 50 miles. 

Under 7 in the column of lat., we find 6.3, which added to the 

45.2 miles, make 51.5 miles as the total northings for the run of 
b1 miles. 

In the same manner, we find under 5 in the dep. column, 2.14. 
Tlemoving the decimal point one place to the right, and we have 
21.4 miles as the dep. or eastings due to 50 miles. Under 7 in the 
dep. column, we find 2.9 miles, which added to the 21.4, wc have 

24.3 miles as the total dep. or eastings due to 57 miles on that 
course. 



PLANE SAILING. 



43 



N. 


S. 


E. 


W. 




21.7 
52.9 

38.8 


2.6 


40.6 

9.7 




113.4 


2.6 


50.3 

2.6 

47.7 



2 Kequired the diff. lat. and dep. for the following courses 
and distances, viz.: 



1. S. W. by W.=S. 5 p. W. 46 miles, 

2. S. iE. =S. ip. E. 53 '' 

3. S.byW.JW.=S.lJp.W.40 '' 
Ans. Diff. lat. or southings=113.4 miles. 

Dep. or westings = 47.7 '^ 



Let the student prepare a table with a column for each of the 
components and a line for each of the courses, as in the example. 

Having taken the components for each of the several courses 
from the traverse table and arranged them in their respective 
columns, add the numbers in each of the several columns, 
writing their sum at the foot of the column. 

Then the difference latitude will be the algebraic sum of the 
northings and southings, and 

The departure will be the algebraic sum of the eastings and 
westings, whence. 

Take the difference of the northings and southings for the dif- 
ference of latitude, and 

Take the difference of the eastings and westings for departure. 

Thus, in the example, the sum of the southings is 113.4 miles, 
and as there are no northings, this is all difference of latitude. 

And the difference between the eastings and the westings is 
found to be 47.7 miles westings. 

Note. — The table of Natural Sines and Cosines, elsewhere 
explained, is a traverse table for distance unity for all courses 
varying by *V from up to 90°. The column of cosines corres- 
ponding to difference of latitude, and the column of sine, to 
departure. This table should be used when precision in regard to 
the course is wanted. 

The tabular components of the course, being multiplied by the 
distance, gives the actual components. 

Rhumb Line. — In consequence of the convergence of the mer- 
idian, as we approach the pole, or recede from it, in sailing on any 

*The intervals of our table vary by 5'. 



44 A MANUAL OF NAVIGATION FOR THE LAKES. 

oblique course, the ship makes a curved line or track, called a 
Rhumb Line. The course of ship is frequently called a Rhumb. 
This curve is always convex toward the equator. 

The constancy of the angle, or course, in crossing the several 
meridians, while they are all inclined to each other, except on the 
equator, is what gives curvature to the course, and it is this prop- 
erty of the rhumb that makes the results of plane sailing rigor- 
ourly correct, notwithstanding a spherical surface cannot be 
developed on a plane. 

Difference of Latitude, is the distance between two parallels 
of latitude. 

As given from a course, by means of the traverse table, it is 
miles or minutes and is called northings or southings, accordingly 
as ship has made northings or southings. But when reduced to 
degrees by dividing by 60 or 69 J, it is called Difference of Latitude. 

Difference of Longitude of two places, is the arc of the 
equator intercepted by the two meridians that pass through the 
two points or places. 

Longitude, is distance in arc (degrees) measured on the equator, 
east or west, from the zero meridian (Greenwich). 

Reduction of Departure to Longitude. — It will be observed 
that eastings or westings called Departure, as found by the pre- 
ceding article, is really difference of longitude, — it being the 
distance between two meridians, — though on a small circle instead 
of the equator. 

And Avhen w^e recollect that there are just as many degrees in a 
small circle as in a large one, we see that some reduction must be 
made before we can measure longitude on a small circle w^ith the 
same unit (the sea mile) that is used in measuring it on a large 
circle. 

Departure, though strictly difference of longitude, is not called 
such till this reduction is made. 

Two methods of converting departure to difference of longitude 
are the following: 

First. We may reduce the measuring unit in the same propor- 
tion as that by which the small circle has been reduced from the 
large one; or 

Second. We may expand or enlarge the departure in the same 
proportion as that by which the equator is greater than the parallel, 



PLANE SAILING. 45 

SO as to make it embrace as many degrees on the equator as it 

otherwise would on the parallel. 

Bearing in mind that the circumferences of circles vary as their 

diameters, and that the cosine of any latitude is the radius of that 

parallel, we have the following proportion: 

-r>. J .. J length of any arc \ .. f length of correspond- 

.cos. .. ^ on the equator / " \ ing arc on the parallel. 

Then, bearing in mind that R is unity, and multiplying extremes 
and means, we have the 

Arc of any parallels r'^^ri^Tr'^'"? equatorial arc 
^xv; ^x .* J pc* u V. (^ multiplied by cosine of parallel. 

The use of this equation is seen in the following problem: 
Required the length of a degree of longitude in any latitude, 
L, say 43°, that on the equator being 60 miles. 
Ans. Arc=60Xcos. 43° 

=60X7314=43.88 miles. 
Or suppose we wieh to find the number of feet in 1^ of longitude 
on parallel 43°, we would have to multiply the equatorial minute, 
0087 feet, by the cosine of 43°, thus, 

5087X.7314=4452 feet. Ans. 
Thus we see that any unit of measure for longitude on the 
equator, may be used for measuring longitude on any parallel, by 
first multiplying that unit by the cosine of the parallel. 

Second method. By inverting the terms of the proportion in 
the preceding article, we have, 

p J Tf f ^^^ ^^^ ^^ ^^^ 1 ^ corresponding arc on 
L.OS. i.: J^:: | parallel of L. / ' t the equator. 

Whence, 

-o f any arc on the ) j ( corresponding arc on 

^ parallel of L. ^ * / the equator. 



Or, 

Cos. L I parallel of L. f \ the equator. 



R { any arc on the } ^^ ( corresponding arc on 



But, 

=Secant. 

Cos. 

"Whence, we have the following Rule for converting departure 

into difference of longitude, viz.: 

Multiply the departure into the secant of the latitude. 



46 A MANUAL OF NAVIGATION FOR THE LAKES. 

Example: How much longitude will 60 miles of departure 
embrace on parallel L=40°, 60°, and 80°? 
Solution: 

Secant 40°=1.305, then 60X1.305= 78.3 miles. ) 

'' 60°=2.000, " 60X2.000=120.0 '' V Answers. 
<i 80°=5.759, '' 60X5.759=345.5 '' ) 

The student will observe that the longitude or difference of 
longitude found above, is minutes, which must be divided by 60 
to reduce them to degrees. 

We could divide by the cosine of the L and get the same results, 
but it is easier to multiply by secant. 

Middle Latitude Sailing. — The methods of plane sailing, 
though correct in their results, are incomplete in that they do not 
determine the particular parallel on which departure shall be 
reduced to difference of longitude, but, 

Middle Latitude Sailing has for its object to determine on 
what paralled the departure corresponding to any rhumb -line, 
shall be converted into diference of longitude. 

The methods of finding the components of a course, is identical 
with that for Plane Sailing, and indeed, for all the sailings. 

It is the practice among seamen, to assume the middle paralled 
of any rhumb-line, as the parallel on which to make this reduc- 
tion. This is well for short runs, but only roughly approximate 
for small courses, in high latitudes, with large distances. 

The true parallel, on which to make this reduction is somewhat 
above the middle of latitude, — just how much is shown by a table 
of corrections given at page 76 of Bowditch's Navigation, called 
Workman's Table. The [construction of this table is fully 
explained in Prof. Coffin^s Navigation, Problems 5 to 10 of Plane 
Sailing, but too abstruse for an elementary work like this. 

It is table XI. of this work. 

Preparatory to the discussion and solution of problems, it is 
well to introduce and explain some notation, for the purpose of 
shortening our work. 

C=the course. L==latitude left. 

d=the distance. L^ ^latitude arrived at. 

I =the difference of latitude. p=departure. 

D=the difference of longitude. L^^corrected middle latitude. 

Thus, in the triangle ABC. 

C B being the meridian, the side. 



MIDDLE LATITUDE SAILING. 



47 




A C=the distance d. 

C B=the difference latitude I , 

A B=the departure p. 

And when the departure is expanded from A B into m n, it is 
called difference of longitude and represented hy D. 

Let the student make himself familiar with this notation, as it 
will help him greatly in getting clear apprehension of his work. 

Example: Sailing from L^65° N on a 
course 0=^. 42° E, distance d=648 miles. 
Required difference of latitude I and 
difference of longitude D. 

Let the student construct the problem 
carefully to scale. — constructing C by its 
chord. 

Solution: As in Plane Sailing, find 
difference of latitude and departure. 

Fig. 20. 

1. Dep. p=dXsin C (=42°)=648X.6691=433.6^ or miles. 

2. Diff. lat. I =dXcos C (=42°)=648X.7431=481.5^ or miles. 

3. Or in degrees =481.5^60=8°. IJ^ 

4. Lat. attained, Li=65°-f-8°.li^=73°.lJ^ 

5. Middle lat. =(65°+73Mi)--2=69°.f 

6. Correction to Middle lat. (see table) = 16^ 

7. Corrected middle lat. L2=69°.|+16^=Say 69°.17^ 

8. Then, by the rule of the preceeding article 
Diff. of long. D=Sec. L^ Xp, or 

2.8263X433.6=1225.5^ or 20°.25J^=D. 
Thus we have our answer in equation (2) and in equation (8.) 
1=481.2 miles, or 8°.1J^ 
D= 20°.25}^ 

Let the student go carefully over this work, looking all the 
numbers out of the tables and performing all the indicated work. 
By this means he will very quickly gain command of the 
problems. 

The student will observe that in equation (4) we added the 
difference of latitude to L to find L^ . If we had made southings 
instead of northings, with our course, we should have used the — 
instead of the + sign. 



48 A MANUAL OF NAVIGATION FOR THE LAKES. 

Instead of multiplying the departure p. by the see. of L, we 
could have devided by cosine of L^ thus: 

433.G--.3538=1225.5^ or 20°.25J^ as before. 

A table for reducing departure to difference of longitude is 
given (table X.) that will give results slightly less than the pre- 
ceeding solution, yet more nearly correct, as it is adapted to 
the periodical or actual form of the earth, rather than that of the 
sphere, as is the table of cosines. 

It is calculated for parallels 30^ apart, from to 80°. When 
it is desired to be more precise, the divisor for internediate 
minutes may be readily interporlated. The divisors will be found 
to be slightly greater than the cosines of the corresponding 
parallels. Thus in the preceeding case, the divisor for L2= 
69°,17^=.355, instead of .3538 and gives D=20°,21J^ instead of 
20°,25J^ 

In using this table, we have merely to divide the departure by 
the value of one minute for the parallel, as given by the table. 

Parallel Sailing, is the finding of the place of ship where it 
sails on a parallel of latitude. 

In this case, the entire distance is departure, the converting of 
which into difference of longitude is already explained, and there- 
fore requires no further attention. 

Mercator's Sailing*. — To devise a means of representing cor- 
rectly, large areas of the earth's surface, and at the same time to 
simplify the reduction of departure to difference of longitude, 
Girard Mercator, in 1566, invented a chart called 3Iercator's 
Chart. This chart, from its many merits, has come into general 
use, the world over, by maritime people. 

To construct this chart, he first expanded all the parallels as we 
have done, thereby making the meridianal distance on all parallels 
equal between any two meridians, and thereby making the mer- 
idians all parallel instead of convergent toward the poles, as on 
the sphere. Then, to preserve the relative positions and the rela- 
tive magnitude of objects, he expanded the meridians in the same 
proportion. 

Each minute of the meridian, from the equator up to the limit 
of the chart, — usually 80° of latitude, — was multiplied by the 
secant of its middle latitude, precisely as we expanded our depart- 
ure to difference of longitude (see page 47) and the sum of these 
augmented minutes, up to any parallel, was called the meridianal 



MERCATOR S SAILING. 



49 



parts for that parallel,—!, e., the expanded meridian. Thus, 
the actual distance from the equator to the parallel 42° would be, 

42X^0=2520 miles, geographic, 
but when each mile is multiplied by the secant of its middle lati- 
tude and added into one sum, we have for parallel 42°, 2782 mer- 
idianal parts. 

To distinguish between the two measurements, the expanded 
distance, on the meridian lines is called Augmented Latitude, 
while that on the sphere, as usually measured, is called Proper 
Latitude. 

The difference of two augmented latitudes is called Meridianal 
Difference of Latitude. Thus, the meridianal parts for 42° 
(abbreviated to M. P.) is 2781.7. The M. P. for 44° is 2945 8, and 
their difference is 2945.8 — 2781 .7=164.1 , which is the meridianal 
difference of latitude between 42° and 44°, whereas their proper 
difference of latitude would be but 120. But, if the two parallels 
be on opposite sides of the equator, then their sum is the merid- 
ianal difference of latitude. 

The course on a Mercator's chart is represented by a straight 
line. This is a consequence of parallel meridians. 

The relation of the parts is seen in the following figure: 



Difi Long 



In the triangle ABC, 
CB=Proper Diff. Latitude =1 . 
CA=Distance =d, 

BA=Departure =p, 

C ] =Augmented Diff. Latitude=m. 

a 1 =Augmented Departure ) -p. 

=^Dift'erence of Longitude ) * 

Course=angle ACB =C. 

From the properties of similar 
triangles, we have, 

CI '.1 a::R:tan. C, whence, 
] aX-R-=Cl Xtan. C, 
but \ a is diff. long. D; R is unity 
and c 1 is the augmented difference 
of latitude, whence, 

D=c] Xtan. C=m tan. C. 
That is to say. 

The difference of longitude is found by multiplying the aug- 
mented difference of latitude by the tangent of the course. 




50 A MANUAL OF NAVIGATION FOR THE LAKES. 

Thus we find the difference of longitude without the necessity 
of first finding the departure. This is one of the advantages of 
Mercator's sailing. But we cannot measure distances by scale on 
this chart. 

Problem: Given the course C=X 26° E, the distance d=142 
miles, L=28° N. Required L^ and D. 

Solution: Preparatory to finding L^ , we must find 1, as in plane 
sailing. Then, augmenting 1, we have m. 

lr=-dXcos, C=142X.8988=rl27.6^=2°, 7.6^ 
Li=-L+1 =28+2°, 7.6^=30°, 7.6^ Ans. 

Meridianal parts (M. P.) of Li=1897.1 

M. P. of L =1751.2 



Meridianal difference of lat.=m=145.9'' Then, 
Diff. long. D=mXtan. C (=26°) 

=145.9X.488=71.2^=1°. 11.2^ Ans. 
Second Solution. Working this example by middle latitude 
sailing, will show the agreement of the two methods. 
As in plane sailing, 

1=2°, 7.6^ and L^ =30°, 7.6^ 

Middle lat.=(L+Li)--2=(28°+30°, 7.6^^2=29°, 3.8^ 
Correction to middle latitude (see table) == V 

Whence, L-^ =29°, 5^ say, 

p==dXsin. C=142X.4384 = 62.24 miles. 

D=pXsecant L^ (=29°, 50=62.24^X1.144= 71.2^ 

= 1°, 11.2^ Ans. 
Or the same result as before. 

The student must keep fresh in mind the notation of pages 46 
and 47. 

Note. — As the tables of sines, tangents, etc., given in this work, 
vary by 5^ of arc, the results obtainable with them will not cor- 
respond strictly with the results given here. 

Problem: Given two places by their geographical co-ordinates 
of place. 

L =46°, 10^ S. Long. 46°, 30^ E. 

Li=52°, 15^ S. Long. 51°, 10^ E. 
Required C and d from L to Li. 

Solution: From the co-ordinates of place Ave have the base and 
perpendicular of a right triangle from which to find the hypoth- 



mercator's sailing, 51 

enuse, which is the d, and the angle opposite the departure, 

which is C. 

M. P. of L 1 (=52°, 150 =3689.6 

M. P. of L (=46°, 100 =3130.0 



Meridian difference of latitude=ni = 559.6 miles. 

Diff. long, D=(51°, 10^—46°, 300=4°, 40^=280 miles. 
By equation 2, page 9, 

Tan. C=D^-meridian difference of latitude 
'' **=280^-^559.6=.5003=tan. 26°, 35^ Ans. 
Proper difference of lat.=(52°, 15^—46°, 100=6°, 5^=365 miles. 
Distance d=] Xsecant C (=26°. 350 

=365X1.1182=408.1 miles. Ans. 

. / C=S. 26°, 35^ E. 
^^s- \ d=408.1 miles. 
It will be observed that d is found from the proper difference 
of latitude 1. This is necessary because oblique distances cannot 
be measured on Mercator's chart. 

Solution by middle latitude sailing: 
Proper difference of latitude=52°, 15^—46°, 10^=6°, 5^=365=1. 
Middle latitude=(52°, 15^+46°, 100-^2=49°, 12 J^ 
Correction for middle latitude (see table)=5^. 
Corrected middle latitude L=2=49°, 17J^. 

Differenceof longitude D=(51°, 10^—46°, 30^)=4°, 40^=280 miles. 
Departure p=cos. L^XD. 

==.6522X280=182.6 miles. 
Tan. C=p^l=182.6^365=.5003=tan. 26°, 35^ Answer, 
as before. Also, 
Distance d=lXsec. C (=26°, 350 

=365X1.1182=408.1 miles. Ans., as before. 
Examples for exercise: 

1. Sailed from L=42°, 30^ N., and long. 58°, 51^ W. S. W. by 
S., 591 miles. Required the latitude and longitude in. 

Ans. Latitude 34°, 19^ N. 

Longitude 65°, 51^ W. 

2. A ship sailed from L=49°, 57^ N., and long. 30°, 00^ W., 
on course C=S. 39° W., till she arrives at Li=45°, 31^ N. Re- 
quired the distance sailed and the longitude in. 

Ans, Distance=342.3 miles. 

Longitude in=35'^, 21^ W. 



52 



A MANUAL OF NAVIGATION FOR THE LAKES. 



Let the student construct these problems to scale, carefully, 
before attempting numerical solution. 

Traverse Sailing, — When a vessel sails on a number of courses 
in making a run, she is said to make a traverse or irregular track. 
And the finding of the equivalent of the traverse in one course 
and distance, is called traverse sailing. While the operation or 
the solution of the question is called the reduction of the tra- 
verse, or finding the course and distance "made good," or reduc- 
ing a day's work. 

For moderate distances, the method of plane sailing is satis- 
factory. But on longer voyages, some one of the other sailings 
must be combined with it for the purpose of keeping account of 
the longitude. It is one of the most useful of the several 
msthods. An example will illustrate: 

Sailed N. N. E.=N. 2 p. E., 140 miles, 

Thence, N. E. by E.=X. 5 p. E., 48 '' 
Thence, east, - - 26 ^* 

Thence, S. by W.=S. 1 p. W., 36 '' 
Thence, S.S. W. J W.=S. 3J p. W\, 76 '' 
Required the equivalent sing-Ie course and distan^^e, or course 
and distance "made good." 

First, prepare a table in which to arrange the several courses, 
with their distances and their several components, — providing a 
column also for changing the courses into degrees, as follows: 





Course. 


Degrees. 


Dist. 


North' gs 


South'gs 


Eastings 


W^est'gs. 


1 


N. 2p. E. 


22°, 30^ 


140 


.924 

129.4 




.383 

53.6 




2 


N. 5 p. E. 


56°, Id' 


48 


.555 

26.6 




.831 

39.9 




3 


East. 


90°, 00^ 


26 






1.000 

26.0 




4 


S. 1 p. W. 


11°, 15^ 


36 




.981 

35.3 




.195 

7.0 


5 


S.2Jp.W. 


28°, 07^ 


76 




.882 

67.0 




.471 
35.8 






156.0 


102.3 


119.5 


42.8 






102.3 




42.8 






53.7 


76.7 





Having arranged the courses in order, with their degrees and 
distances in their proper columns, take the cosines of the courses 

from a table of natural sines and cosines, and write them in the 



TRAVERSE SAILING. 53 

upper left hand of the space for the northings or southings for 
that course, as the case may be. 

And write the sines of the courses in the column for departure 
— eastings or Avestings, as the case may be; thus, 

In the upper left hand of the space for northings, is written the 
cosine of 22°, 30^; and in the column for eastings, is written the 
sine of 22°, 30^ 

These are the factors with which to multiply the distance, 140 
miles, for the components of course; and so for all the courses. 

Having multiplied each distance into the cosine and sine of its 
course, and arranged the products in their respective columns, 
write the sum of the components of each column at the foot of the 
same, and 

Find the difference of the northings and southings and the dif- 
ference of the eastings and westings, thus, 

In this case, we find the northings in excess of the southings 
53.7 miles; and the eastings exceed the westings 76.7 miles. 

Thus far, the problem is merely another form of plane sailing. 
We have now our difference of latitude 1=53.7 miles, and depart- 
ure 76.7 miles, from which to find the course C and the distance 
d, whence, 

p-Hl=76.7--54.7=1.4021=tan. 54°, 30^=course, 
and as the course takes its name from its components, we have 
the course, 

X. 54°, 30^ E„ or X. E. J E., 
and multiplying northings by secant of C, we have, 
Distance=54.7Xl.T22=94.2 miles. 

Instead of taking our components from a traverse table, we 
have taken the cosines and sines to distance unity, and multiplied 
by their respective distances. 

This is for the purpose of showing a more precise way of com- 
puting the components of a course, as the ordinary traverse tables 
are not computed for anything lower than degrees, — though this 
is not designed to supercede the use of the traverse table for 
ordinary work. 

If, now, the question of longitude is involved, we must introduce 
middle latitude sailing, as in the following 



54 



A MANUAL OF NAVIGATION FOR THE LAKES. 



Example: From latitude 41°, 12^ N., longitude 20° W., make 
the following traverse: Required the latitude and longitude 
attained, and the course and distance ^'made good," 

1. S. W. by W. 21 miles. 2. S. W. J S, 31 miles. 
3. W.S.W.JS. 16 '^ 4. S. IE. 18 '' 

5. S. W. i W. 14 '^ 6. W. J N. 30 '' 

As in the former case, rule seven columns for courses, etc. 



Angle 


Dist. 


North 


gs 


South 'gs 


Eastings 


AVest'gs. 


1. 


S.W.byW. 


=5 p. 


21 






11.7 




17.5 


2. 


s. w. ^ s. 


=3^ p. 


31 






24.0 




19.7 


3. 


W.S.W.JS. 


=5Jp. 


16 






7.i> 




14.1 


4. 


S. f E. 


= fp. 


18 






17.8 


2.6 




o. 


s. w. 1 ^y. 


=4ip. 


14 






9.4 




10.4 


6. 


W. J N. 


==7J p. 


30 


2.9 








29.8 










2.y 




70.4 
2.9 


2.6 


91.5 

2.6 




Southi 


ngs^67.5 


Westings 


=88.9 



We find we have made difference of latitude southings 67.5 
miles, and departure westings 88.9 miles. Dividing the southings 
by 60, we have, 

67.5-r-60=l°, 7Y southings in latilude, or, 
latitude attained is 

41°, 12^— 1.7J=40°, 4y ^.=LK 
Then, dividing departure p by difference of latitude 1, we have, 

p^l^88.9--67.5=rl.k7=tan. 52°, 48^=course, 
and because the components of our course are southings and 
westings, the name of our course is S. 52°, 48'' W. 

Multiplying difference of latitude by secant of C, we have dis- 
tance d=67.5Xl-655=111.7 miles. 
By middle latitude sailing, we have, 

D=pXsec. L 2=88.9X1.318=117.2 miles, 

=1°, 57^ diff. long, westings. 
Then the longitude in 

=20°+l°, 57^=21°, 57^ W. 
Collecting our results, we have, 

Li=40°, 4J^N. 
C=S. 52°, 48^ W. 
d=111.7 miles. 
Longitude in=21°, 57^ W. 
It will be observed that w^e have in this problem taken half the 
sum of the extreme latitudes for the middle latitude L^. This is 



CURRENT SAILING. 55 

the usual practice for small distances. But for large distances, or 
with small C, it is necessary to apply the correction of table XI. 

Or the following rule may be used for finding the parallel on 
which the departure p must be augmented for difference of 
longitude. 

Take the parallel whose cosine is half the sum of the cosines of 
the extreme parallels, for the middle parallel. (This rule is 
original). 

This, although not rigorously correct, is practically so for all 
ordinary cases. It gives the L^ very slightly too large for L less 
than 45°, and slightly too small for L larger than 45°. 

The student should solve all his questions by at least two meth- 
ods, as a means of checking against mistakes, — one of which 
should be by construction, for fixing the problem clearly in 
the mind; another is by inspection for some of the parts. 

Thus, after we have found the 1 and p for a traverse, we can 
search the latitude and departure columns of a traverse table till 
we find these two components in the same line. The corres- 
ponding distance will be found in the ^'distance" column on the 
left, and the course will be found at head or foot of page or 
column. 

But this plan, though rigorously correct in principle, will not 
generally be found satisfactory. 

The traverse tables are not computed generally to arcs varying 
by less than one degree, so that the two components can rarely be 
found precisely, — hence some interpolation will be required. 

A better method is to divide both the components by the dif- 
ference of latitude 1, Then, by a principle of trigonometry, we 
have the tangent of the course C, which may be found from the 
table of Natural Sines, Tangents, etc. And from the same table, 
the secant of C is found, which gives us the distance d for unity. 
Then, multiplying sec. C by 1, we have the d desired, as in the 
preceding examples. 

Current Sailing. — The effect of a current on a vessel is the 
same as that of another course and distance, the course being the 
direction of the current, and the distance being the rate per unit 
of time, — as an hour, — multiplied by the time sailed in the same. 

If a vessel sails with a current, she will be ahead of her reck- 
oning, by the amount of the motion of the current, during the 



oG 



A MANUAL OF NAVIGATION FOR THE LAKES. 



time of sailing in it. And if she sail againfet the current, she will 
be behind her reckoning by the same amount. 

If she sail across the current, she will be carried with it through 
the distance the current moves while the ship is in it. 

The direction of the current, with regard to the meridian, is 
called the set ; and the rate at which it runs per hour is called 
the drift, 

From the above conditions, it is seen that in all cases, when 
sailing with a current, the set and drift must be regarded as an 
independent course and distance. Also that time is an element 
to be considered. 

Example: A ship made the following traverse in a current 
setting X. by W., at the rate of two miles per hour. 

1. S. W. J W., 2 hours, 8 miles per hour=16 miles, 

2. W. JS. 3^-7^^ '' —21 " 

3. W. byN. 3^-6*^ '^ =18 '' 
Required the course and distance ^'made good.'^ 







Dist. 


X. 


S. 


E. 


W. 


S. \Y. J W 


50°, 37^ 

84°, 22^ 
78°, 45^ 
11°, 15^ 


16 
21 

18 
16 


3.5 
15.7 


10.1 

1.1 




12.4 


W. JS 


"^O.d 


W. bvN 


17.6 


N.byW 


3.1 






Current 






19.2 
11.2 


11.2 




n^A) 




miles. 



-= 8.0 miles. 
Course=]S'. 81°, 34^ W. Distance, 54.7 miles. 

Example 2: A ship sails S. 17° E. for two hours, at two miles 
per hour, as indicated by ship's log; thence S. 18° W. four hours, 
at the rate of seven miles per hour; and during the whole time, 
the current sets X. 76°, at the rate of two miles per hour. Required 
the course and distance ''made good." 

Ans. C=S. 21°, 49^ W. Distance==42 miles. 

Note. — It will be observed that this is merely the application 
of traverse sailing, with an extra course and distance introduced 
into the traverse. 

Oblique Sailing. — It will be observed that, up to the present, 
all our determinations for place of ship have been made by means 
of the properties of the right plane triangle. Some questions 
require the use of the properties of oblique triangles. The 



OBLIQUE SAILING. 



57 



finding of place of ship by this means, or the determining of ques- 
tions in navigation by this means, is called oblique sailillg*. 

It is used chiefly in the survey of harbors, — in the location of 
shoals, with regard to their bearing from other objects, — or in 
finding compass errors. A few examples will illustrate. 

But these questions will involve the use of some means on board 
ship, and as but very few of our lake vessels are provided with 
such appliances, I give the following method for the 

Construction ^'f the Dumb Card, which should be on every 
ship on the lakes: 

Let the ship's carpenter describe a circle, say six or eight feet 
in diameter, with 
center C in center 
line of ship. 

Through C, and 
at right angles to 
center line, draw the 
line 0. 

Divide each J of 
the circle into eight 
equal parts, and by 
means of chalk-line 
or straight-edge, 
transfer these points 
onto the rail, and 
make a clean, deep 
mark at the inter- 
section. Number these marks from to 4, from forward and from 
abeam, on each side of the head of ship; and number from to 
1, 2, 3, etc., points abaft the beam. 

Let the marks on the rail be numbered with the number of the 
point, by driving brass or tin headed nails to indicate the number. 

Let the center of the circle be marked, as by driving some nails 
at the intersection of the fore-and-aft and 'thwart-ship lines. 

Then, to use this card, the eye being over the center and look- 
ing over the rail, the bearing of any object from ship is readily 
seen; and the bearing is referred to the nearest zero line, — 
thus we would say an object appears IJ points abaft the port 




Fig. 22. 



58 A MANUAL OF NAVIGATION FOR THE LAKES. 

beam, 2 points forward of the starboard beam, or 3 points off the 
port bow, as the case may be. 

Problem: A ship-master being about to sail, wishes to examine 
his compass, as to its accuracy. He observes from his chart that 
when 30 miles out on his proposed voyage, he will be 4 miles to 
the left of a certain lighthouse that stands on a headland. When 
Hearing the light, he observed it to be If points forward of the 
starboard beam. After sailing 6 miles, as indicated by the log, 
and on the same course, the light appeared IJ points abaft the 
beam. Having shaped his course, on the supposition that his 
compass was correct, he wishes to know from the above observa- 
tions if it is so. If not, which way is it out, and how much. 

Solution: Construct the problem as per figure. Draw any right 
line AB to represent course of ship, and on it take AB=6 miles, 
to any convenient scale, At A draw 
the line AC, forward of the beam If p. 
And at B,*draw BC, abaft the beam IJ 
p. They w^ill intersect at C. 

It will be observed that each of the 

angles B and A, in the triangle ABC, 

is the compliment of the observed angle, 

i, e., the angle at A=8 p. — If p.=6J p., 

^•* etc. 

Then, bearing in mind that the sum 
of the three angles of any plane triangle 
^^' is 16 p., we have only to take the com- 

pliment of the sums of the angles A and B, to know C=3J p. 

Then, in the triangle ABC, knowing one side and the three 
angles, we have a case for the **sine proportion,' ' by which 
we have, 

Sin. C:AB::sin. B : AC 

: : sin. A : BC, or 

Sin. 3J p. (=.5958) :6 miles :: sin. 6 J p. (=.9569): AC (=9.63 miles) 

:: sin. 6J p. (=.9416):BC (=9.48 miles). 

See table V, for the sines, etc., of points and parts, and let the 
student verify by careful construction, and let him perform the 
numerical work here indicated. 




UBLIQUJ 



59 



Our question requires us to know the height CD of the triangle. 
This can be measured by scale, or computed numerically. 

By equation 3, page 9, 

CD==ACXsin. A {=6^ p.) 
3=9.63X.9-A16=9.06 miles. 

By the conditions of our question, we should be 4 miles to the 
left of C, but the above work shows us to be 9.06 miles to left, — 
that is to say, our compass has taken us to left of our true course 
O.06 miles in 30. Then, by equation 2, page 9, we have, 

Tan. course=5--30=.1666=tan. 9°, 29^ 
So that our compass is out to the left 9J°=f p. 

In the preceding solution, it would have been sufficient to make 
one proportion for one side of the triangle ABC; but tinding both 
sides, gave means of checking our work. Thus, either side, AC 
or BC, multiplied into the sine of its adjacent angle, should pro- 
duce the perpendicular CD=9.06 miles. 

Problem. The following problem is of the same character as 
the preceding, except the bearings are referred to the meridian 
instead of to center line of \ 

ship: 

A port, n, bears X. E. J 
]N". from a port, m. At 26 
miles out from m, toward 
B, directly abreast the port 
beam, is a light, distant 6 
miles, when the ship is on 
the right course. The mas- 
ter having shaped his 
course on the supposition 
that his compass was cor- 
rect, when nearing the 
light, took its bearing, N. 
by W. i W.--N. IJ p. W. 
After sailing on same 
course 8 miles further, as ^^' 

indicated by ship's log, he took a second bearing, W. by X. | N. 
^^N. 6J p. W. Eequired to know the compass error, if any, — 
liow much and which way. 

Solution by Construction: Draw any line, as AE, through the 
page, vertically, for the meridian. From any point on this line, 




60 A MANUAL OF NAVIGATION FOR THE LAKES. 

as at A, set off the course of ship, 3J p. to right of the meridian 
and 8 miles long, to any convenient scale. From the same point 
A, set off the bearing of the light C, IJ p. to left of meridian. 
From B set off BC 6J p. to the left of the meridian, or, Avhat is 
the same thing, make the angle ABC=to the supplement of the 
bearings AB and AC, viz., 6J p, Then by scale, we find, 
CD=7.8 miles. Answer. 
Solution by Oblique Trigonometry: In the triangle ABC, the 
Angle at A=e5 p. or 3 J -^1 J, 
Angle at B=6} p. or 16 p— (3J+6}), 
Angle at C=4f p. or 16 p. — (5-f 6}). 
Then, by the sine proportion: 

Sin. C: AB :: sin. A : BC, 

:: sin. B : AC, or, 
Sin. 4J p. (=.8032) : 8 :: sin. ^ p. (=.8315) : BC (=8.28 miles), 
:: sin. 6} p. (—.9416): AC (=9.38 miles). 
Then, either side, AC or BC, multiplied by the sine of its adjacent 
angle, gives the perpendicular, thus, 

AC (=9.38 miles)Xsin. 5 p. (=.8315)=CD (=7.79 miles), 
and BC (=8.28 miles)Xsin. 6i p. (=.9416)=CD (=7.79 miles). 

But, by the conditions of our problem, CD should be 6 miles^ 
that is to say, our compass has taken us to the right, say 1.8 miles 
in 26, which, by right trigonometry, corresponds to an angle 
of 4°. 

Answer. Compass out to the right 4°. 

Many other questions could be proposed for solution by oblique 
trigonometry, but they would be of a class that seldom or never 
occur in practice, — are more curious than useful, — so we spend no 
time with them. Following are a few miscellaneous questions: 

Distance of an Obiect by two Bearings. — The two preceding 
problems are at the foundation of table XII. They are deemed 
of so much importance to ship masters who have occasion to 
round headlands in the night, that a table has been prepared from 
which the distance of a light, or other object from a ship, — as also 
the line of the ship's course, at right angles from the light or 
object, may be told in advance, or before reaching the vicinity of 
the '^danger line.'' 

But the table here presented is quite different from that used at 
sea, and for the following reason: There, the bearings of the 



OBLIQUE SAILING. 61 

object or light are referred to the meridian, by means of the stand- 
ard compass, which is furnished with a movable ring and sights 
for the purpose. 

But, on the lakes, our compasses being boxed up in the pilot 
house, are not available for such work, even if they had the ring 
and sights. We can only take the bearing of the object from the 
ship's center line, by means of the ship's dumb compass, — reading 
the bearing directly from the card, when she is fortunate enough 
to have one, — which, indeed, is the better way. 

The table is constructed generally by solving a number of tri- 
angles, varying in their angles by J or J point, through such 
limits as would embrace all the cases likely to occur, both for the 
side opposite the first bearing and for the perpendicular distance 
of the base or line of the ship's bearing, produced, from the 
light or object, for a distance of unity. 

The results are tabulated in column under the first bearings 
taken, and in line of the second bearings as factors by which to 
multiply the distance run between the times of observation for the 
distances sought, — the larger product giving the distance from 
ship to light at the time of the second observation; the smaller 
product giving the distance to light or object when it comes 
abeam, — or the height of the triangle, as it is technically termed. 

In the table which I give (table XII) I introduce the factor, as 
above, for the distance of the light from ship at the time of second 
observation. But, instead of the second factor, I introduce the 
sine of the second bearing, to be multiplied by distance of ship 
from light, for the perpendicular distance of object, or height of 
the triangle. 

An example will illustrate the use of the table: 

Being about to pass a headland in the night, the track by which, 
as given by the chart, lies two miles to the right, and not knowing 
whether my compass is correct or whether I was in the proper 
track when shaping course, I wish to know if my course will take 
me the proper distance to the right of the light. Soon after 
making the light, I found it to bear 2 points to the left of the 
ship's heading. After sailing 3J miles, as indicated by ship's log, 
it bore 4J points to left. Eequired the distance of ship from the 
light at time of second observation, also the perpendicular dis- 
tance of the track of ship from the light. (See Fig. 25). 



62 



A MANUAL OF NAVIGATION FOR THE LAKES. 



Solution: Draw the indefinite line A m to represent the ship's 
course. From any point A in this line, draw the line AC two 
points to the left of A m, and from A set off to B the distance 
sailed, 3 J miles. From B draw BC 4 J p. to the left of ship's 
course, meeting AC in C. Then is C the place of the light. 

From table XII, in column of 2 p. and in line of 4J p., will be 
found the decimal .81. This, multiplied by the distance, 3J miles, 
gives us BC. the distance of light from ship=2.83 miles. Thia 

again multiplied by the sine of 4J p. 
(=:.773)=2.19 miles=DC, the dis- 
tance of ship's track from the light. 
This problem is also available in 
finding compass errors. 

Course aud Distance from the 
Co-ordinates of Place. — The won- 
derful clearness and fullness of 
detail in our lake charts are pur- 
chased at the expense of one con- 
venience, and that is the finding of 
course and distance, in some cases. 

The scale of the chart is so large 

as to make it impracticable to give 

us a full or entire lake on one sheet; 

as a consequence, they are given in 

sections, which must be combined 

before courses can be obtained from. 

Fig. 25. them in all cases. 

This combination is inconvenient at times, nevertheless the course 

between ports represented on different sections can be readily 

obtained from their *^ co-ordinates of place," as given in the list 

of lights. An example w^ill illustrate: 

At Sturgeon Bay Canal. Required the course and distance to 
Michigan City. 

Solution: From the U. S. List of Lights for the Lakes, we find 
Sturgeon Bay in latitude 44°, 47^, longitude 87°, 18^. Michigan 
City in latitude 41°, 43^ longitude 86°, 54^ 

The difference of latitude is 3°, 04^=212 miles, 
The difference of longitude is 24^=20 miles, 
for the mean latitude 43°, 15^. 
Whence, in sailing froni Sturgeon Bay to Michigan City, we must 




OBLIQUE SAILING. 



63 



^•0 



'/ZF 



make southings 212 miles, and eastings 20 miles. The southings 

are found by multiplying the difference of latitude in degrees by 

the value of 1 degree=69.15 miles, giving 212 

miles. The eastings are found by multiplying 

the difference of longitude, 24^ by the value 

of 1 minute of longitude, for the mean 

latitude, as given by table X, giving us 20 

miles. 

Constructing a triangle, as in Fig. 26, 

with base 212 and with perpendicular 20, and 

measuring the angle at A with protractor 

or scale of chart, we find the angle say \ p. 

That is to sav, our course is S. J E. 


Then, by plane trigonometry, multiplying 

the base 212 by the secant of course (^=1.0048, 

table III), we have 213 miles for distance. 

By plane trigonometry, also, the course 
may be found, for 

20--212=.0943=tan. 5°, 25^=say \ p. 
This, it must be remembered, is the chart 
course, which must be modified by variation, 
— in this case \ p. to the right, — making the 
magnetic course S. f E.; and this must 
again be modified by deviation, if any, also by 
leeway and for current, if any. 

Fig. 26. 



CHAPTEK ly. 

Construction of Charts. 

Construction and Use of Mercator's Chart. — The principles 
underlying the construction of this chart have been examined 
under the head of Mercator's Sailing. It only remains to make 
the practical application. 

Suppose we were to construct a Mercator's chart of the territory 
embracing the great lakes, — say from latitude 40° to 50°, and 
from longitude 75° to 95° W. We must prepare the skeleton or 
blank, as follows: (See table on following page). 

1. Write down in a column the degrees and parts of a degree 
that are required to be represented on the sides of the map, — as 
the whole degrees, J° or the ^°. (See table IX). 

2. From a table of meridianal parts (abbreviated to M. P.) 
write the M. P's corresponding to each degree and part of a 
degree, in an adjoining column. 

3. From the M. P's of the highest latitude, subtract those of 
the lowest latitude. Thus, the M. P's of 50° (--3474.5), the M. P's 
of 40° (=2622.7), their difference is 891.8, which is the depth of 
our map in latitude, in the scale units that rejDresent 1°- 

4. From the M. P's of 50°, take the M. P's of the whole 
degrees, 49°, 48°, 47°, etc., setting down the differences opposite 
their respective latitudes. Thus each difference will represent the 
distance between the consecutive parallels that are one degree 
apart, and the place of each parallel can be marked on the side of 
the map at one placing of the scale, after we 

5. Add the several differences consecutively, for the height 
up to the consecutive parallels. Thus, 

The M. P's for 50°=3474.5 
'' '' '' '' 49°=3382.1 



92.4, seen at foot of 3d column, 
M. P. for 49°— M. P. for 48°=90.6, as seen in 3d column, etc. 



MEKCATOR S CHART. 



65 



In this manner will the widths of single degrees be found and 
recorded in the 3d column. Then, adding consecutively, as per 
(5), we have the height of the first parallel from base of map= 
78.9 scale units. Then 78.9-i-80.1=159=height of 42° parallel 
from base of map; 159+81.4=240.4 M. P's for the height of the 
43d parallel, etc. In this manner was the fourth column pro- 
duced, that shows the height of each parallel from base of map. 

6. In the widths for the J degrees, subtract the M. P's of 49J° 
from those of 50°, and we get 46.5 M. P's, found at the foot of 
the 5th column. The M. P's for 49J°— those for 49°=45.9 M. P's, 
seen in the 5th column. And in this manner was the width of 
€ach consecutive half degree found. 

Table of Elements for a Mercator's Chart. 
Latitude 40° to 50°. 



Lat. 


Meridianal 


Widths of 


Heights in 


Widths of 


Parts. 


Degrees. 


Latitude. 


Half Deg. 


40 


2622.7 






39.3 


i 


2662.0 






39.6 


41 


2701.6 


78.9 


78.9 


39.9 


i 


2741.5 






40.2 


42 


2781.7 


80.1 


159.0 


40,6 


1 


2822.3 






40.8 


43 


2863.1 


81.4 


240.4 


41.2 


i 


2904.3 






41.5 


44 


2945.8 


82.7 


323.1 


41.9 


J 


2987.7 






42.3 


45 


3030.0 


84.2 


407.3 


42.6 


i 


3072.6 






43.0 


46 


3115.6 


85.6 


492.8 


43.4 


i 


3159.0 






43.7 


47 


3202.7 


87.1 


580.0 


44.2 


i 


3246.9 






44.6 


48 


3291.5 


88.8 


668.8 


45.1 


1 


3336.6 






45.5 


49 


3382.1 


90.6 


759.4 


45.9 


i 


3428,0 






46,5 


50 


3474.5 


92.4 


851.8 





The M. P's for half degrees were found by taking the difference 
of the consecutive half degrees in columns 1 and 2. 

The table for the values of the degrees of latitude, being pre- 
pared, we are ready to construct the framework or skeleto-n of our 
chart. 



66 A MANUAL OF NAVIGATION FOR THE LAKES. 

1. Assume any convenient scale. — preferably one that is 
decimally divided, and set off on a horizontal line, at the foot of 
the chart (the south side,) the amount of longitude required, — 
in this case, 20° or 120 miles (1 mile being the scale unit), mark- 
ing the points for the meridians, — each degree or each alternate 
degree, as wanted. 

2. At such extremity of the line representing the width of the 
map in longitude, erect a perpendicular for the extreme meridian 
or sides of the chart. 

3. On each of these meridians, set up the distances given in 
the preceding table, for the places of the several latitudes, com- 
mencing at the bottom of the sheet, thus. 

Set up 78.9 M. P's for the place of the 41° parallel, 

u 1590 u u u a 490 

a 240.4 '' '' " '' 43° '' etc. 

Thus we have all the parallels located at their proper places, to 
represent the augmented latitude. 

4. We can now locate the places for the half degrees from 
their scale values, in the column for the widths of half degrees. 

The framework or skeleton is now ready to take the location of 
places, — as towns, coast-lines, rivers, islands, boundaries, etc.,. 
which are located from their known places of latitude and long- 
itude, by means of two T squares, — one locating the latitude, the 
other the longitude, — and their intersection being marked by a 
needle-point or sharp pencil. Thus, a number of points in the 
boundary of a lake or bay being located and a fair line traced 
from one to another, the shore line is located, etc. 

It is to be remarked that the "meridiaual parts" given in our 
table IX, are those that have been in use a long time, as first con- 
structed, on the supposition that the earth is a sphere. But more 
modern w^orks compute them for the earth regarded as a spheroid^ 
— making them slightly smaller than for the sphere, for any given 
latitude. 

Bearings and Distances. — The bearing* between any two 
points on a Mercator's chart, is very readily found. 

It is only necessary to draw a straight line between the two 
points and apply a protractor to any meridian crossed by this 
line, to read the bearing. This, and the straight ''rhumb line,'' 
are two of its conveniencies. But not so with distance. This 
cannot be measured by scale, on a Mercator's chart, except in 



PLANE CHART. 67 

the direction of longitude; and this measurement must be multi- 
plied by the cosine of the latitude before it is available for use, 
for it will be remembered that the whole map, away from the 
equator, has been expanded. 

Scale measurements caiiiiot be made ill any oblique direction, 

because the scale varies from the equator toward the pole. Thus, 
at 40° of latitude, a half degree is only 39.3 M. P., while at 50°, 
it is 46.5 M. P., — and this is an objection. 

Construction of the Plane Chart. — In the plane or rectang- 
ular chart, the meridians are parallel lines at a uniform distance 
apart. 

This distance, for one degree, is found by multiplying the 
equatorial distance, 60 miles geographic or 69.15 statute, by the 
cosine of the latitude for which the map is made. 

The following considerations will show the amount of error for 
such a chart: 

First. It must be remembered that the meridians at the equa- 
tor are parallel, and at the poles they have their maximum inclin- 
ation, which is the whole difference of longitude, and that between 
these limits, the inclination varies as the sine of the latitude. 

Example: Suppose we wish to make a rectangular or plane 
chart for the latitude of 42°, for an area of one or two counties, 
or say 30 miles square. Multiplying 60, the number of geograph- 
ical miles in one degree, by the sine of 42° (=.6691) we have 40.14 
geographic miles for one degree of longitude, or 46.28 statute 
miles, and the error resulting from their being parallel, would be 
60^Xsin. 42° (=.6691)=40^ for 1° of longitude, i. e., in a block 
30 miles square, the south side would be about 55 rods too Small, 
while on the north it would be that amount too large. 

The Conical Projection. — In the conical projection, the mer- 
idians are all right lines, but they are inclined by the amount due 
to the central latitude of the map. There are two methods for 
this projection, — the tangent and the secant. The latter being 
the more accurate, is the one we will illustrate. 

In Fig. 27, let ABC be the arc of latitude to be embraced by 
the map. Set off from each end of the arc of latitude, one fourth 
of its length, to a and a^, and through these points draw the 
secant intersecting the earth's axis produced in D. Then, the 
cone, of which aa^ D is an element, will, when rolled out or 



A MANUAL OF NAVIGATION FOR THE LAKES. 



developed, be a portion of a circle and the radials through it, and 
of which the secant aa^ D is one, will represent the meridians of 
the sphere, and circles described through it from D as a center, 
will be the parallels of latitude. The points whose latitude is a or 
a^, would be correctly represented in magnitude, while those near 
A and C would be too large and those near B would be represented 
too small. 

The following example will illustrate, viz., to construct the lines 
for a map embracing latitude 40° to 50°, and 75° to 95° of long- 
itude, i. e., the vicinity of the great lakes. 
Dividing the difference of latitude, 10°, 
into four parts ,we find the two parallels, 
a and ai, to be five degrees apart, and two 
and one-half degrees from the extremity 
of the map; and a=42J°, and ai=47J°. 

The length of one degree of longitude 
on the parallel a 

=cos. 42J (=.7373)X60=44.24 miles, 
and on a^, 

=cos. 47J {=.6756)X60==40.54 miles, 
and their distance apart on the meridians, 

==^X60-=300 miles. 
And the radius of the developed parallel a, 
is the cotangent of 42 J° multiplied by the 
radius of the earth; also that of a^ is co- 
tangent of 47J° multiplied by the earth's 
radius. These being too large to be used 
as sweeps with which to describe the parallels, we must devise 
some other means of sweeping arcs of circles. 




Fig. 27. 



Draw a line vertically through the middle of the map, to 
represent the middle meridian. From head of the map, set 
down 150 miles for the place of the parallel 47J°. Below this, set 
down 300 miles more for the place of the parallel 42J°, and below 
that, set down 150 for the parallel 40°, limiting the south side of 
the map. 

From a, (see Fig. 28) with one or two times the width of one 
degree of longitude, 44.24 miles, as computed above, accordingly 
as a meridian is wanted for every alternate degree, describe the 
arc of a circle, both to right and left. 



ORIENTING SHIP. 



69 



In the same manner, from a^, with the same multiple of the 
width, 40.54 miles, describe an arc on each side of the meridian, 
and through these arcs draw the meridians, as for 83° and 87° of 
longitude. 

With twice theii distances, sweep again from same centers, for 
the meridians 81° and 89°, and three times their distances, sweep 
arcs for 79° and 91°, etc. 

Produce the two outside meridians of the map till they meet 
the middle meridian, from which point sweep the parallels of 
latitude 40°, 42°, 44°, etc. 

The above is the most practical and readily available of the 
many methods in use for representing areas of several hundred 
miles square. A more accurate method, called the PolyCOllic 
Projection, is in use by the U. S. Coast and Geodetic Survey, 
but as it is somewhat abstruse in its construction, we do not illus- 
trate it. 

Orieiitiiig" Ship. — Two methods, the Direct and Eeciprocal, of 
finding compass errors, have been treated, together with the appli- 
ances for the work. But the subject is of such importance, that 
we give a number more, in- 
cluding the Orienting of Ship, 
for the purpose of compen- 
sating ship's compass. 

There are many opportun- 
ities of finding lines of ' 'known 
bearing," as between inter- 
visible lights, or between two 
headlands, or between a light 
and a headland. Such lines 
are said to have a port-hand 
or a starboard-hand bearing, 
accordingly as the line is to the left hand or to the right of the 
meridian, as the observer looks to one of the objects. 

Thus, the line Skilligille-Waugoshanee, bears X. 2J p. E. The 
observer standing in this line and looking either to the north or 
the south, will see one of the lights to the right of the meridian, 
whence, the light is said to have a starboard-hand bearing. 
This notation will be found very convenient for defining the bear- 
ing of lines. 




IS- </d ^/ 



?^ ?s ^0 & 79 71 
Fi^. 28. 



70 



A MANUAL OF NAVIGATION FOR THE LAKES. 



Example: The ship crosses a line known from the chart, to have 
a port-hand bearing of 2 J p. At the moment of crossing, the line 
is seen J p. abaft the port beam, as indicated by the dumb card, 
described on page 57; and the ship's compass indicated ship to be 
headed E. by N. J N., and the chart gave the variation E. J p. 
Required whether the compass is in error, which way and how 
much. 



Solution: Draw two lines, X. S. and E. W., to represent the 
meridian and the prime vertical, at right angles to each other. 
Through the intersection of these lines, draw the reference line, 
to the left of the meridian 2 J p., and draw the magnetic meridian 

J p. to the right of the true mer- 
idian and the E. W. line. Then, 
by observation, the ship's head 
(indicated by the arrow-point) 
was 8^^ p. to the right of the ref- 
erence line. From this, subtract 
2| p., the distance of the mag- 
netic meridian to the right of the 
true meridian, and find ship's 
head to be 5J p. to the right of 
magnetic meridian, leaving 2 J p. 
between ship's head and magnetic 
E.; but the compass shows only 
IJ p., whence compass is out to 
left J p. Thus the dumb card 
gives great facility and clearness 
to this class of questions. 

Example 2: The line St. Hel- 

ena-McGulpin's Pt,, has a port- 

A steamer, in crossing this line, 

The compass indicated 




Fig. 29. 

liand course of 44J°, or say 4 p. 

observed it 6 J p, off the starboard bow. 

ship's head at S. W. by W., variation J p. to the right. Required 

-compass error, if any. 



Solution: As in the former example, draw the cardinal lines 
and the line representing the magnetic meridian. Then, by obser- 
vation, the ship's head was 6J p. to the left of the reference line, 
or lOJ p. to the left of the N., or 5 J p. to the right of the S. But 
hy ship's compass, the ship's head was only 5 p. to the right of 



ORIENTINCi SHIP. 



71 



S., whence, bv deviation and variation, it is J p. out to right 
Deducting variation J p., the needle is out to right f p. 

These two problems will show the method of treating this class 
of questions, and the great utility of the dumb card. 

With the dumb compass, described elsewhere, there would be 
no computation whatever. The sights being set to the bearing of 
the reference line and then turned into that line, the true heading 
of ship is read from the card. 

Orienting Sliip by Amplitudes. — The bearing of the sun at 
sunrise or sunset, when referred to the prime vertical, i. e., to the 
E. or W., is called the amplitude of the sun. 

This angle or bearing depends on the latitude of the observer 
and on the declination of the sun. It is a problem of spher- 
ical trigonometry, and is solved 
by the following equation: 

Sin. amplitude^ 

sec. lat.Xsin. dec, or 
=sin. dec.^-cos, lat. 
Example: In latitude 43° 
N., with sun's declination 21"^ 
^., what is the bearing of the 
sun at sunset or sunrise. 
Solution: 

Sec. L (=43°)=. 1.367 

Sin. D (=21°)= .3584 

5448 
10936 
, 6835 
4101 




Amplitude 



=sin. 29°,- 20^=. 4899308 



Fig. 30. 



As the bearing takes its name from the decimation, the sun will 
bear E. 29°, 20^ N. at sunrise, and W. 29°, 20^ N. at sunset,— or 
better, :N. 60°, 40^ E. and N. 60°, 40^ W. 

The dumb compass makes this reference line available for 
orienting ship, with great facility. Thus, 

1. Set the index to the reading of the amplitude and turn the 
card under the index, to the right or to the left, by the amount 



72 A MANUAL OF NAViaATION FOR THE LAKES. 

of variation for the locality. This reduces the true amplitude to 
a magnetic amplitude, which is what is wanted when we are look- 
ing for compass deviation or when we are compensating for devi- 
ation. . 

An example will best illustrate, though at the risk of some rep- 
etition: 

It IS required to orient ship at Buffalo. Lat. 43° X., sun's dec. 



Solution: From the table of amplitudes (table IV), select the 
amplitude 25°. corresponding to the given latitude and declina- 
ation. The dumb compass being in position, with the zero line 
of the lower index, or lubber line, set toward the head of the ves- 
sel, and the north side of the card approximately to the north, set 
the index of the right bar to read the amplitude, viz., E. 25° N. 
for sunrise, or W. 25° X. for sunset. 

Turn the card under the index 5° to the left. Clamp the index 
and card together and sight to the sun. Then is the card oriented 
to the magnetic meridian. 

Hold the card and sights pointing to the sun and give ship the 
^ 'wheel'' till ship's head is at the desired cardinal point, as indi- 
cated by the reading of the lower index. Then will ship be 
oriented to the magnetic meridian. 

The two preceding methods of finding true bearing of ship's 
course, are eminently practical and of easy attainment, readily 
available and requiring but little or no mathematical work; but 
each has its inconvenience. 

In the first, the ship must recover her place in the reference 
line after each observation, in order to avoid parallax. 

In the second, by amplitudes, according to the usual practice, 
the time for work is too limited. The sun soon leaves the place 
defined by the amplitude, when observations for compass error 
. must cease. But there is a property in the rate of the change of 
the sun's place, by which this inconvenience may be avoided, at 
least for some time, and that is, 

A Constancy in the Rate of Change of the Sun's Azimuth. 

It is found that, with latitude and declination of the same name. 



ORIENTING SHIP. 73 

the rate of change of the sun's azimuth, between sunrise and the 
time of his crossing the prime vertical, is practically constant for 
any latitude above 40". Thus, Avith latitude 37°, declination 0°, 
the rate of change is one degree in 6 minutes, 40 seconds. With 
declination 12°, the rate is 1 degree in 6 minutes, 20 seconds. 
Again, in latitude 45°, with declination 0°, the change of the sun's 
azimuth is 1 degree in 5 minutes, 50 seconds. With declination 
12°, it is 1 degree in 5 minutes, 30 seconds, and with declination 
20°, it is 1 degree in 5 minutes, 30 seconds. And this rate will 
hold for a few minutes after passing the prime vertical, after which 
it increases rapidly. 

This property in the rate of the change of the sun's azimuth, 
between sunrise and the prime vertical, or between the prime 
vertical and sunset, makes it easy to construct a table that is in 
all respects a veritable table of the sun's time azimuth. 

To facilitate this work, I have computed the rate of change for 
each latitude, and written it immediately under the latitude of 
each of the columns of table lY. 

An example wdll best illustrate. With latitude 43° and declin- 
ation 15°, alike, the amplitude 20°, 44^, — say 21°. Observe the 
w^atch time of the sun's rising and change this amplitude to azi- 
muth by taking the compliment. Then, say at 5 A. M., the sun 
rises IS". 69° E.; at 5:06 A. m., lie is at N. 70° E.; at 5:12 a. m., he 
is x^. 71° E., etc., till we get 90° of azimuth, when the sun is ia 
the prime vertical and the rate of change begins to increase. The 
evening w^ould do as well, but then the observer would have to 
know the time of sunset and the error of his watch on apparent 
time. — things not always known. 

By Time Azimuths of the Sun, — The relations of latitude and 
declination, and local apparent time, are such that these elements 
being known, the bearing of the sun can be readily deduced. 
These bearings are called Time Azimuths. 

They are deemed of such importance at sea that tables of them 
for all declinations up to 80°, and for time up to the length of the 
longest day, have been computed. Among the best of these azi- 
muth tables, are those of Major General E. Shortrede, F.K.A.S., 
a part of which we give in this work, with explanations for their 
use. (See table VIj. 



74 A MANUAL. OF NAVIGATION FOR THE LAKES. 

These tables require a precise knowledge of the local apparent 
time, and, as a consequence, at sea, require daily observations for 
that element. 

But with us on the lakes, or along the coast, now that we have 
Standard time established, and the latitude and longitude of all 
the lights along the coast and lakes being known, the case is very 
different. The astronomical work required at sea for local ap- 
parent time, is dispensed with on the lakes, — for the reason that 
it is done for all time, and to a degree of precision that we could 
not hope to attain, — in the IT. S. List of Lights. 

This with the equation of time as furnished in the Nautical 
Almanac, and our Standard or Mean Time, which is now 
available in nearly every R. R. Station, gives us the means of 
knowing the apparent time at any locality, with great precision, 
whence the Azimuth tables are available to us without the 
astronomical instruments, and without the nautical service, 
requisite for their use at sea. 

Local Apparent Time. — As our watches are supposed to be 
regulated to Standard or Mean time for some particular meridian, 
and as we want the local apparent time, it is necessary to find the 
error of our watch on such time. 

And as we are to find this error from the difference of longitude 
between the meridian of the observer, and that to which our 
watch is regulated, the reduction of arc or longitude to time, and 
the reverse, will be involved in the question. (See table VIII.) 

This reduction of Arc to Time, and the reverse is made by 
means of margined tables to the tables of natural Sines Tangents, 
etc. which see. The outer column on the left contains the time 
for the degrees at the top of the page, and the minutes of arc 
on the left. (See large works on Navigation.) 

The outer column on the right, gives the time for the degrees 
at the foot of the page, and the minutee of arc on the right. 

In the time columns the hours under 3, will be found at the 
head of the column. The minutes in black faced figures, and 
the seconds in common figures in the column. 

The hours from 3 to 6 will be found at the foot of the column. 



ORIENTING SHIP. /O 

Thus, for 34°, 28^ of arc, the corresponding time=2 h. 17 m. 52 s. 
Thus, for 76°, 16^ of arc, the corresponding time=5 h. 5 m. 4 s. 

The first step in this problem, is to find that part of the error 
of our watch due to difference of longitude. 

This is found by taking the difference of longitude of different 
places, as given in the List of Lights, on the lakes, and reducing 
the same to time at the rate of one hour for 15° of longitude, one 
minute of time for 15^ of longitude, etc. or 4 m. to 1°, 

Thus the longitude of Chicago is given as 87°, 37^, while the 
longitude of the standard time Avatch is 90°, — the difference of 
longitude is 2°, 23^. This, reduced to time by table XIII, is 9 m, 
32 s. That is to say the difference between standard time, and 
the local mean time of Chicago, is 9 m. 32 s. 

And because the meridian of the standard time 90°, is West oi' 
that of Chicago, the watch is slow, and requires the {-\-,) plus 
correction to reduce it to the local mean time of Chicago. 

Again, — the longitude of Buffalo is given as 78°, 54^, while the 
longitude of the standard time watch, — to which Buffalo mean 
time is referred, is 75°, — and their difference is 3° 54^, which 
reduced to time, is 15 m. 36 s. But in this case, the standard 
meridian is East of the local meridian, — whence the time correc- 
tion must be subtractive to reduce the standard to the local mean 
time, — and the minus sign is prefixed to the time correction. In 
this manner was table VII prepared for most places on the lakes. 

Having found the error of our standard time watch on the 
mean time of any locality, it remains to find the error on ''Appar- 
ent Time," i. e., the error on the time indicated by the sun. 

Apparent time does not progress uniformly as does mean time. 
As a consequence, it is sometimes fast and sometimes slow of the 
mean time clock. Just how much fast or slow is shown by the 
Nautical Almanac, under the heading "Equation of Time," tech- 
nically called the Eq. of time. 

The sign prefixed to this correction of time, — or equation of 
time, — or eq. of time, — is that for reducing apparent time to mean 
time. Thus, when the sun is slow, the eq. of time has the + sign 
prefixed to reduce apparent time to mean time, and when he is 
fast, the — sign is applied. 



76 A MANUAL OF NAVIGATION FOR THE LAKES. 

But in our case, we wish to reduce our mean time to apparent, 
i. e., we wish to find the error of our mean local time on apparent 
local time, whence the above signs must be reversed. Accord- 
ingly, in table V, we have prefixed to the time correction, or eq. 
of time, the sign requisite to reduce mean time to apparent time, 
whence, 

To find the error of the standard time watch on local apparent 
time, we have only to take the algebraic sum of the two errors, or 
corrections of tables V and VTI, observing the signs. 

A few examples will illustrate this problem: 

1. At Chicago, June loth (no matter what year), required the 
error of the standard time watch on local apparent time, for the 
purpose of finding azimuth from the sun. 

Solution : 

Time correction for diff. long, (table VII), + 9 m., 32 s. 
Eq. of time (table Y), - - — 10 s. 



Error of watch on apparent time, - + 9 m., 22 s. 

That is to say, we must count the time forward 9 m., 22 s., on the 
watch, to get apparent local time. 

2. At Chicago, November 1, required the error of standard 
time on local apparent time. 
Solution: 

Time correction for diff. of long, (table YII), + 9m., 32s. 
Eq. of time (table Y), - - - +16m., 20s. 



Error of watch on local apparent time, -^25m., 52s. 

As before, we must read our watch forward, say 26 m., to get local 
apparent time. 

3. At Cleveland, May 24, required the error of the standard 
time watch on local apparent time. 
Solution: 

Time correction for diff. long, (table YII), — 26 m.,48 s. 

Eq. of time (table Y), - - + 3 m., 26 s. 

Error of watch on local apparent time, — 23 m., 22 s. 
Here we must count backwards on our watch to get local appar- 
ent time, 23 m., 22 s. 

It will be observed we have not given the year of the date in 
our examples. This is not necessary for purposes of azimuth, for 



ORIENTING SHIP. 77 

the corrections being the mean corrections, — equations, as they 
are called, — for four years, the error cannot be more than the 
change of declination for J day, and cannot, therefore, appreci- 
ably vitiate the resulting azimuth. 

Explanation of the Tables of Time Azimuths. — The tables 
are arranged with the declinations in a column on the left of the 
page, and azimuths in a line at the head of the column, the hour 
anole corresponding to any declination and azimuth, being found 
in the line of the one and column of the other. 

The declination 0, is written in the middle of the column, and 
it increases upwards and downwards to 24°, varying by intervals 
of 2°. The upper part of the column is marked + to correspond 
Avith latitude of the same name; the lower part is marked — to 
correspond with latitude of different name. 

Definite azimuths are assumed, and the corresponding H. A. 
(hour angles) are computed for each degree of latitude and for 
each second degree of declination, and written in columns under 
their respective azimuths (Z), and in lines opposite their respect- 
ive declinations, the degrees of the latitude being written at the 
head of the page. 

The table commenees with the sun at apparent noon, giving on 
the first page, asknuths (Z) for intervals of 5°, and on the second 
page, at intervals of 3°. 

Azimuths, as well as declinations, might have been given at 
smaller intervals, but these intervals were assumed to keep the 
tables from being too voluminous, while they are yet small enough 
to m.ake interpolation easy. The folloA\dng definitions will help 
to a more ready apprehension of the manner of using the tables : 

Upper and Lower Pole. — The upper pole is that one having 
the same name as the latitude of the observer. Thus, with us, the 
north pole is the upper pole, while the south is the lower pole. 

North and South Meridian. — That part of the observer's 
meridian between himself and the north pole, is called the north 
meridian, while that part between himself and the south pole, is 
called the south meridian. The meridian opposite in longitude, 
is called the nether meridian. 

Supplement of H. A., is what the hour angle lacks of 12 hours, 
or 180°. 



78 A MANUAL OF NAVIGATION FOR THE LAKES. 

P. M. and A. M, — In popular language, these initials iraplv the 
after part or the fore part of the day. But in this question, A.M. 
implies that part of the forenoon within 90° of the meridian; 
while P. M. implies that part of the afternoon within 90° of the 
meridian. 

Morning', is that part of the forenoon between sunrise and the 
prime vertical, or 90° from no on; and Evening is that part of the 
day between the prime vertical and sunset. These terms do not 
occur, except with latitude and declination of the same name. 

Azimuths are reckoned from that pole that is on the same side 
of the prime vertical as the sun. Thus, with the stin in N. declin- 
ation, we reckon azimuth from the north, morning* and evening^ 
i. e., before the sun passes the prime vertical in the morning, and 
after he passes it in the P. M. 

Zenith and ]?fadir. — Zenith is the pole of the observer's hori- 
zon, while Xadir is the pole of the nether horizon. So that when 
the sun is more than six hours from the meridian of the observer, 
it is less than six hours from the nether meridian, and conse- 
quently above the nether horizon; and the supplement for the H. 
A. for the evening is the H. A. for the morning. 

As the terms latitude, azimuth, declination and H. A , do not, in 
themselves, show immediately when the sun is above or below the 
horizon, this information is given by the l)Iack line in the several 
columns. The terms below the black line belong to the sun when 
below the horizon, and as a consequence, to the evening and the 
morning. 

To Use the Tables. — The tables answer such questions as the 
following : 

At what time in the day will the sun be at a given azimuth Z 
from the observer ? Or, at a given time in the day, what is the 
bearing of the sun? In reply to the latter question, we will say : 

If the latitude and declination are of the same name, i. e., both 
I^. or both S.. find the declination in the upper part of the page, 
on the left, with the latitude at the head of the page. Then fol- 
low the line of the declination till the H. A., or time nearest to it, 
is found. Then, over the column containing the H. A. thus found, 
will be the azimuth Z. 

1. Thus, in latitude 43, with sun in declination 20°, both N., 
what will be the bearing Z of the sun at 34 m. P. M.? 



ORIENTING SHIP. /^ 

The latitude and declination being of the same name, we l,ook 
for (he declination on the upper left of the page, with the given 
lat tude at the head. Then, following the line till we find the 
given H. A., 34 m., and looking to the head of the column, we 
find Z=20°. That is to say, the sun will then bear S. 20° W. Or 
at 11 h., 2G m. A. M., the sun would bear S. 20° E. 

2. In latitude 46°, with declination 18°, both N., what will be 
the bearing of the sun at 7 o'clock in the morning? 

Taking up the declination in the upper left of the page and fol- 
lowing the line of H. A. to the right, we find that the sun does 
not cross the prime vertical till it is 4 h., 56 m., from noon, so the 
sun is still on the morning side of the prime vertical. Looking ia 
the line of 16° declination, in the lower part of the page, we find 
at 7 h,, 04 m., the sun is in the prime vertical, i. e., due east. 

3. What will be the bearing of the sun at 8 o'clock A. M., at 
10 and 11, apparent time? 

As the H. A. in these are the times from noon, we must look 
for the supplement of A. M. time. Thus, at H, A. 4 h., 08 m., 

Z=81°; for H. A. 2 h., 01 m., Z=50°; for H. A. 1 h., 06 m., Z=rrr30°. 

With the preceding general information concerning the table, 

the following more concise instructions, adapted to our particular 

latitude, viz,, the coast and lakes of the U, S., will be permissable: 

To And the H. A. for Morning', with the sun in X. declina- 
tion. Look in line of the declination for the given latitude, and 
under the black line, where will be found the H. A. from sunrise 
to the prime meridian, with the corresponding Z at the head of 
the column. 

The H. A. will be found increasing, as also the Z, from left to 
right, till the sun crosses the prime vertical. At the crossing, the 
morning changes to A. M., and the supplement of the clock time 
becomes the H. A. which is found in the line of the declination 
in the upper part of the page, then 

The A. M. H. A's will be found in the upper part of the page, 
with the corresponding Z at the head of the column, both the Z 
and TLA. diminishing from right to left, from prime vertical to 
noon, where they vanish. 

KoTE.— It must not be forgotten that while the sun is north of 
the prime vertical, the Z must be measured from the north mer- 



80 A MANUAL OF NAVIGATION FOR THE LAKES. 

idian, or, if read from the south meridian, the Z will be found in 
the corresponding column at foot of page. 

Thus, for latitude 43° N.. declination 20° N., in the morning at 
4 h., 46 m., apparent time, the sun is 63° from the north meridian, 
i. e., its Z is N. 63° E. Ail h., 32 m., it is due east, i. e., it is in 
the prime vertical, and its Z is 90°, as seen both at the foot and 
the head of the right hand column. The supplement of this time, 
or 4 h., 28 m., shows the same thing, in the same column, in the 
upper part of the page, in line of the given declination, 20°. 

With the Sun in South Declination, we shall never have use 
for the H. A's under the black line. 

The supplement of the clock time will be the H. A. from sun- 
rise to noon, where it is 0. 

These commence in the line of the given declination, imme- 
diately to the left of the black line, and diminish to the left, — as 
do also the Z's. 

In the p. 31., the clock time is the H. A., which will increase 
from left to right, — from noon to sunset, — which will be found at 
the black line or immediately above it. 

Example: Thus, with latitude 43° N. and declination 8° S. the 
last P. M. H. A. given is 5 h., 24 m., with corresponding Z=78°. 
The H. A. and Z would be the same for A. M., but the clock time 
would be the supplement of the H. A., or 6 h., 36 m. 

Interpolation, — We may want Z for intervals, or at times 
intermediate between those given in the table. These are easily 
obtained. 

The difference between two consecutive H. A. in the same line, 
is the difference or interval between the two corresponding Z's. 
This, divided by the difference of the Z's, gives the time due to a 
change of 1° in the Z. 

Thus, for latitude 36°, declination 20°, both + the H. A. for 
Z=60, is 1 h., 38 m. For Z=63°, the H. A. 1 h.^47 m., i. e., in 
9 m. Z has changed 3°, or 1° in 3 m. 

"Whence, having found the H. A. for Z, we may set the index 
of our dumb compass, or azimuth ring forward 1 degree for each 
3 minutes, or for each 4 minutes, or 5 minutes, as the case may be. 
By forward, is meant the motion from left to right, as we read the 
dial of a watch, — supposing the eye of observer at the center of 
card. 

Interpolation for Declination. — This is done with the same 
facility. Thus in latitude 43°, declination 19°, with names alike, 



ORIENTING SHIP. 81 

or +. The H. A. for 60° azimuth is required. Here, the H. A. 
for declination 20° is 2 h., 10 m. For declination 22°, the H. A. 
is 2 h., 02 m. The difference is 8 m., whence the. H. A. corres- 
ponding to declination i9° is 2 h., 6 m., for Z:==60°. This can all 
be done mentally. • 

The rate of change of azimuth is not alike in all parts of the 
day. It is slowest at the prime vertical and most rapid at the 
meridian, as before shown. 

Preparation in Advance. — The ship-master or compass- 
adjuster who contemplates to swing ship for compass errors, par- 
ticularly in the A. M. part of the day, should find the error of his 
watch on local apparent time in advance. Take the supplement 
of the H. A. in advance and reduce them to his Avatch time, — 
thereby saving time by being ready for any favorable moment, 
and avoiding liability to mistake by not having too many things 
on the mind at once. 

A convenient way of interpolating is to find the rate at the azi- 
muth, Z changes, between two consecutive H. A., as given in the 
tables, precisely as given in the methods by amplitudes (page 71). 
Thus, with latitude 43° N., declination 18° N., the 
H. A. for Z=66° is 2 h., 40 m. 
'' '' ^'=69° is 2 h., 52 m. 



3° 12 m., or Z changes 1° in 4 m. 

H.A. for Z=15° is 27 m. 
'' '' ''=20^ is 37 m. 



b^ 10 m., or Z changes 1° in 2 m., etc. 

The error of the watch on local apparent time being found, and 

the H. A. for any particular Z, or azimuth, being looked out of 

the tables, the work of orienting ship is precisely that explained 

in former articles. 

The method of orienting ship by means of time 
azimuths will cost the student more study, more diligent and 
patient application than any other of the several methods enumer- 
ated for orienting ship, but it is also the most useful, the most 
satisfactory and the most generally available; and now that we 
have standard time established, it is peculiarly adapted to use on 
the lakes and coast, where ii is available without the astronomical 
work requisite for its use at sea. 



CHAPTEK V. 

Terrestrial Magnetism and the Magnetism of Iron in 

Vessels, as Affecting the Compass Needle; and 

THE Correction of Co3ipass Errors. 

Of magnetism we know little more than that it is one of the 
many imponderable agents by which the material universe is act- 
uated. Some of the laws governing its mode of action have been 
discovered by observation. 

Prof. Bartlett, formerly of the Military Academy of West Point, 
gives us to understand that what is called Terrestrial 3Iagnetisiii 
is generated by a thermal wave constantly flowing from east to 
west, caused by the constant change in temperature of that portion 
of the earth's surface exposed to the sun's rays, together with the 
axial motion of the earth. 

And electricians tell us that one of the properties of a magnet- 
ized needle is to take position directly across or at right angles to 
a current of electricity, when brought in proximity to such current. 

As a consequence, if this current were strictly parallel w^ith the 
geographical equator, the magnetic needle w^ould coincide with 
the geographical meridian and there would be no variation, but 
unfortunately, this is not the case. 

For some reason, supposed to be the nonsymmetry of the topo- 
graphical conditions contiguous along the track or to inequality 
of conducting power of different parts of the earth's surface, this 
current is diverted from a parallelism with the earth's equator, 



MAGNETIC EQUATOR. b6 

— resulting in what is called the 'S^ariation" of the needle, — the 
current being sometimes to the right, and again to the left of the 
equator, in its motion from east to west, 

A little to the west of Africa, in longitude about 0, this current, 
or rather the center of this current, crosses the equator, coming 
from the north and taking a southwesterly direction, touches 
South America near latitude 18° S. From this point in its west- 
erly course, it tends gradually northward, reaching the equator 
near longitude 70° W. from Greenwich, and giving westerly var- 
iation in the Indian and Atlantic Oceans, Africa, most of Europe, 
and that part of North America east of the great lakes, wtth a 
small portion of eastern South America. And easterly variation 
to the entire Pacilic Ocean, most of the two Americas, and Asia. 

But this terrestrial magnetism is not the only force that is active 
in giving position to the needle. The magnetism in the iron of 
the ship is the principal disturbing element that makes trouble tor 
navigators, with regard to their compasses. 

Now that the practice is becoming general, on the lakes, of 
compensating ships' compasses for deviation, by the use of oppos- 
ing magnets, it is important that those who are to do this work 
should have a little idea of the manner this local magnetism is 
deduced from terrestrial magnetisms, and of some of the laws of 
its action. Accordingly I offer the following few elementary facts 
and definitions : 

Magnetic Equator. — There is a point in the current spoken 
of, in which the magnetic needle assumes a horizontal position 
when suspended in the magnetic meridian. To the north of this 
point the north end of the needle dips, i. e., inclmes downward, 
while to the south, the south end dips. The line on which the 
needle is horizontal is called the magnetic equator, inconsequence 
of proximity to, and analogy to the geographical equator. The 
Dip is the angle of inclination to the plane of the horizon and 
increases from the magnetic equator to the magnetic pole, each 
way, — but at an irregular rate, — when it is vertical, or 90°. 

Magnetic Latitude, is distance either north or south, defined 
by the dip of the locality. Thus, along the south shore of Lake 
Erie, or the south part of Lake Michigan, the north end of the 
unbalanced magnetized needle would dip about 73°. These local- 
ities would then be said to be in 73° of magnetic latitude, though 
their geographic latitude is only about 42° north. 



84 A MANUAL OF NAVIGATION FOR THE LAKES. 

Line of Force. — The opposite poles of a magnet being at its 
extremities, the direction which the needle takes shows the direc- 
tion of the action of the forces residing in the two ends, or oppo- 
site poles. — such direction being called the line of force. 

It must not be supposed that, as between the forces of the two 
opposite poles, there is anj force of translation, but merely that 
of direction. The maximum intensity of magnetism in any locality 
being in this line of force. 

Neutral Plane. — If a plane be conceived to intersect the line 
of force at right angles, it will show the position of minimum 
force for that locality, or rather the position of no magnetic force. 
It is in this plane that magnetic forces change sign in passing from 
one ])ole to the other, — whence it has come to be called the neu- 
tral ])lane. 

If any object susceptible to the earth's influence, as a bar of 
soft iron, make any angle with the plane, the intensity of the 
magnetic action will be proportioned to the sine of this angle. 
Thus, at the points before mentioned, in latitude 43°, if a bar 
of soft iron be laid in a horizontal position and in the plane of 
the magnet!,, meridian, it will be 17° from the neutral plane. 
Then as the sine of 17° is .2923, it will be acted on by less than .3 
of the earth's magnetism at that place. 

If the rod is up-ended to the vertical, it will be 73° from the 
plane and will therefore be acted upon by .956 of the earth's 
magnet force, — this being the sine of 73°. While if the rod be 
placed horizontally in an east and west line, it will be in the neu- 
tral plane, and therefore will not be acted on at all. This variable 
intensity of the earth's magnetic action makes great trouble for 
navigators. 

Components of Magnetic Force. — The needles in use in all 
compasses are horizontal, while, as we have seen, the earth's 
magnetic force acts in an inclined direction. It is therefore 
necessary to regard the earth's magnetic force as the resultant of 
two other forces, — one horizontal and one perpendicular, or 
rather, vertical. 

The Horizontal Component, is that part of the earth's mag- 
netic force that orients the needle, while the vertical component 
acts constantly in a vertical direction, in the magnetic meridian, 
— varying the dip with the change of magnetic latitude, and, as 
we shall see hereafter, indirectly disturbing the needle. 



INDUCED MAGNETISM. 85 

The Relatire Magnitude of these two forces is shown on 
Plate II, Figs, 2 and 3, bv a plan which I have devised from a 
chart by Mr. R. J. Evans. 

Fig. 2 shows the varying intensity of the total force between the 
magnetic equator and the magnetic pole, for the meridian of about 
170° W. and passing near Behring's Strait, — that at the equator 
being unity. 

It is known to be about 2.3 times as great at the magnetic pole 
as at the equator, — and between these limits it has been found to 
vary nearly as the square root of 1 plus three times the square of 
the sine of the magnetic latitude. 

The curve bounding the section of Fig. 2 is formed by the pre- 
ceding law, and shows that the intensity of the earth's magnetism 
increases faster on first leaving the equator than when nearing the 
pole. 

Fig. 3 represents the corresponding horizontal component. 
This is a maximum at the equator, where the whole force is hor- 
izontal. At the magnetic pole, where the line of force is vertical, 
this horizontal component vanishes. Between these limits it varies 
as the total force at the locality multiplied by the cosine of the 
dip. 

The curve bounding the section of Fig. 3, is made by the above 
proportion, — the result showing that for some distance from the 
equator, toward the pole, the horizontal or directive component 
holds its own with but little diminution, after which it falls off 
rapidly on nearing the pole, where it vanishes. 

Those who have read Dr. Kane's report of his expedition in the 
Arctic Ocean, will remember the trouble he had with his compass 
needle, for want of directive force. 

Fig. 4 represents the Vertical Component. This is nothing 
at the magnetic equator, where the needle is horizontal, and has 
its maximum, — which is the total force, — at the magnetic pole, 
where the unbalanced magnetic neeedle stands vertical. Between 
these limits it varies as the total force of any locality multiplied 
by the sine of the dip for that locality. 

Induced Magnetism. — In consequence of the constant stream 
of magnetism flowing around the earth, all fuliginous matter 
within its influence becomes charged with magnetism by induction. 
Whence, in our (north) latitude, the lower portions of all masses 
of iron, such as columns, vertical shafts, smoke funnels, vertical 



S6 A MANUAL OF NAVIGATION FOR THE LAKES. 

rods of any kind, — being below the neutral plane, — are on the 
polar side of that plane and therefore have the same magnetism 
that is in the north end of the needle. While the upper parts 
being on the equatorial side of this plane, have that kind of mag- 
netism that is in the south end of the needle. 

Hard and Soft Iron. — The influence of this induced magnetism 
on different kinds of iron, is seen in different effects on the com- 
pass needle. 

Hard Iron is slow to receive magnetism by induction, and as 
slow to part with it. If a bar of hard iron, having its longer 
dimension approximately parallel with the ^Mine of force," be 
violently treated, as by bending, twisting, chipping, filing, or in 
putting lathe-work upon it, its magnetism will not only be 
increased, but comparatively fixed. 

Soft Iron on the contrary, becomes magnetic almost imme- 
diately when its longer dimension is placed in a position approx- 
imately parallel with the *^ line of force." It also has its intensity 
increased and comparatively fixed by violent treatment. 

Fixed or Permanent Magnetism, is that which remains con- 
stant, or practically so, in the body to which it pertains, as in a 
hardened steel magnet, or in the so-called loadstone. Magnetism 
in vessels is not permanent. Hard iron retains its magnetism some 
time, but not so permanently as steel. 

The plates of vessels, boilers, smoke funnels, tubes, etc., being 
worked cold, then hammered in riveting and caulking, have their 
magnetism partially fixed, producing a magnetism called sub- 
permanent, i. e., composed partly of permanent and partly of 
transient magnetism. This transient part uf mechanism passes 
away rapidly from a new vessel, by use, making much trouble for 
the navigator. 

Different Effects of the Different Magnetisms.— The differ- 
ence m the effects produced by a fixed magnet, as that of a bar of 
hard steel magnetized, and that produced by a bar of soft iron 
that is made magnetic by induction, is seen in the following 
experiment: 

Apply the end of a soft iron rod to a compass needle, the rod 
being in a level position, at the same height of the compass and 
in an east and west direction. The rod, in such position, being 
in the neutral plane, is not acted upon by the earth's magnetism, 
and as a consequence, cannot affect the needle. !Now, up-end the 



INDUCED MAGNETISM. 87 

rod to a vertical position, keeping one end,— say the lower end, — 
on a level with the compass. The lower end of the rod, in this 
position, is below the neutral plane, and therefore is charged by 
induction with the magnetism that is in the north end of the 
needle, as will be shown by its repelling the north end of the 
needle. 

If now, we lower the rod away, canting it till the upper end 
comes below the level of the compass, — still keeping it vertical, — 
we shall bring the south magnetism of the rod to act on the 
needle, which will be shown by its attraction of the needle, thus 
giving the opposite result, and showing that it is not any pecu- 
liarity of the rod that produces this change, but the almost 
instantaneous effects of the earth's magnetism on the rod. 

Effect of Direction of Ship's Head, while Building". — It has 
been seen from observation that iron ships receive a permanent 
magnetic character from the direction in which the head is located 
during the building. In a vessel built with head to the magnetic 
north, the magnetic meridian would coincide with the ship's plane 
symmetry, — or with her center line, as it is called. 

The forward lower part of the vessel, being below the neutral 
plane (in north latitude), would have that kind of magnetism that 
is in the north end of the needle, while the after end of the ves- 
sel would have south magnetism, — and so for other directions of 
the ship's head. 

Semi-circular Deviation. — In swinging ship, the fixed mag- 
netism of the head iron retains its relative position with the other 
forces and is therefore on one side of the compass needle in one- 
half of the revolution, and on the other side during the other 
half, thereby producing deviation through 180° of azimuth on 
each side of the magnetic meridian, — whence the name semi- 
circular deviation. 

The deviation due to the effects of hard iron, is far from being 
constant. It falls off rapidly with the use of the vessel during the 
first few months of service. As a consequence, a new iron ship 
cannot be depended on lo carry her adjustments long, without 
some change. 

The magnetism resulting from vertical component, being of 
constant sign and constant intensity in the same latitude, also 
produces a semi-circular deviation. And this is always combined 
with that produced by the hard iron, and is made less in amount 
than the latter. 



88 A MANUAI. OF NAVIGATIOX FOR THE LAKES. 

These two parts, which always come together by addition, have 
different properties. 

That depending on hard iron is not constant in its intensity, 
because of the inability of the iron to hold its magnetism perman- 
ently. Hence the term, ^'sub-permanent magnetism." While 
that depending on the vertical component of induced magnetism, 
is constant in the same latitude. 

Quadrant al Deyiatioii, is the result of magnetic action 
induced by the horizontal component in the soft iron in the ship 
and it is constant in amount, or nearly so, — being proportional to 
horizontal component of the locality. And, as will be presently 
seen, it changes sign in each quadrant. Commencing in the first 
quadrant, N. to E., the signs are -j-, — , -f-j — j whence the name 
**quadrantal deviation." 

Graphical Illustration. — The several parts of which devia- 
tion is composed, are illustrated by Fig. 1, of Plate II, 

The two segmental areas, 4-B, — B, represent that part of the 
semi -circular deviation due to hard iron, and which is generally 
the larger part. 

The curve bounding this arc, is a curye of sines, i. e., if the 
deviation at east or west be multiplied by the sine of the azimuth 
of any point in the curve, measured from the Kode of the curve, 
or the place where the curve crosses the magnetic meridian, the 
product w^ill be the deviation at that point. 

Thus, if a deviation at E. be 20°, that for X. N. E. would be 
(see table III) 

sin. 2 p. (=r.384)X20°= 7.7^; and for X. E. 
would be sin. 4 p. (==.707)X20°=14.1°; and for E. N. E. 
would be sin. 6 p. (-=.924)X20°=18.5°. 

These distances set off from the magnetic mei^idian, will give 
points in the curve for the semi-circular deviation. 

The Lnnular Seg-ments, +C, — C, show the part of the semi- 
circular deviation due to the effects of the vertical component. 
The boundry of this area is also a curve of sines. 

The four lunular segments, -j-D, — D, +D, — D, show how 
quadrantal deviation is combined with semi-circular deviation to 
produce the total error of the compass. These, also, are curves 
of sines. 

The Law of Change of Signs will be seen from the following 
considerations: 



QUADKANTAL DEVIATION. 89 

When the head of the ship is to the magnetic north, the mag- 
netism of the ship and that of the earth conspire in one direction 
on the needle, so that, though the directive force be increased, 
there is no deviation, — whence the magnetic north is a nodal 
point of this curve, or nearly so. 

When the ship's head is east magnetic, the horizontal com- 
ponent is in the neutral plane and thus does not act, and as a con- 
sequence, there is no deviation, — whence, the east is also a nodal 
point for this curve. 

In the same manner the south and west may be shown to be 
nodal points. 

The several parts of deviation above enumerated, are called 
*'Co-efflcieiits." They are A, B, C, D, E^, of which we have 
shown B, C and D. 

*'A" is that part of compass error due to what may be called 
Index error, as when the zero line of compass is not parallel with 
center line of ship or when the card is not cemented onto the 
needle in the proper place, — and is usually too small to notice. 

**E" is a small error remaining in the octants after the quad- 
ranted deviation D, has been corrected. It is seldom seen except 
with very long compass needles, and is generally so small as not 
to be worth noticing. 

The preceding illustrations are on the supposition that the sev- 
eral causes of disturbance are symmetrically arranged to each 
other on the two sides of the ship's plane of symmetry, called also 
her ''midship plane;" and that the vertical magnetic plane that 
coincided with the plane of the magnetic meridian during the 
building of ship, coincides also with this 'midship plane, — and 
that the ship's compass is in this plane. But all these conditions 
seldom or never prevail at the same time. 

And this trouble is aggravated by the habit of masters and 
builders of placing their compasses to one side of the center line 
of ship, 

The result is, these several parts of deviation, which always 
come together by addition, algebraically, are shifted in azimuth 
with regard to each other, so that the nodes of the curve are not 
at the magnetic cardinal points, as they otherwise would be, 
nearly. 

This is illustrated in the deviation curves of the ''Trident" and 
"Warrior," Plate I. 



90 A MANUAL OF NAVIGATION FOR THE LAKES. 

In the case of the ''Trident,'^ it will be seen that the curve 
crosses the magnetic meridian at N. J E. and at S. ^ W. — thus 
making one semi-circle f ])oint longer than the other. 

The curve of the ^' Warrior," crosses the meridian at N. 3 p. W. 
and at S. f p. E., thus making the distance between nodes, 2f p. 
greater on one side than on the other; — which makes it impossible 
to bring the needle to place at both North and South or East and 
West as the case might be, by means of opposing magnets. And 
this often brings the compass adjuster into disrepute by people 
who are ignorant of the above facts, and who think that because 
he cannot get good ** reversals" in all cases, that therefore he 
does not know his business. 

A Yivid llliistratioii of the effect of combining quadrantal^ 
with semi-circular deviation, is seen also in the deviation curves 
of the "Trident" and '^Warrior" above referred to. 

It will be remembered that the signs of B and C, when west, 
are — , and that the sign of D is +, — , +, — , commencing in 
the first quadrant, and going around by the East, South and 
West, to the North. 

Constructing the curve bounding the two parts B and C, (as on 
Page 88), — taking the deviation at West, 23°, as the height of 
the curve, we find for intervals of 2 p. of azimuth, the height of 
€urve as follows, viz: (See table III.) 

Sin. 2 p. (=.383) = 8°.8. 

Sin. 4 p. (=.707) X23°=16°.3. 
Sin. 6 p. (=.924) =21°.3. 

Setting off these distances, by means of the scale at head or 
foot of the plate, we have the place of the curve shown by the 
dotted line, thus showing the area embraced by the dotted or 
broken line, to be reduced in the first quadrant, and making the 
deviation curve slightly concave at N. E. 

At S. E. the deviation curve is sharply convex, because the 
two signs being alike ( — ), their parts come together by addition. 

At S. W. the two signs being alike (+), show the deviation 
curve to be sharply convex. 

At the N. W. the two signs being unlike, (Xj — )> the two parts 
come together by subtraction, and we have a lean concave curve, 
as in the first quadrant. Thus: 

The total deviation in the two northern quadrants, is greater 
than the deviation in their diagonally opposite quadrants by 
twice the mean quadrantal deviation of those quadrants. 



HEELING DEVIATION. 91 

The above property affords a ready means of separating quad- 
rantal deviation from the total deviation. Eule: Take half the 
arithmetical difference of the diagonally opposite quadrantal 
deviations, for the mean quadrantal deviation. 

And in constructing the curve, observe that the heights, or 
ordinates of the curve vary as the sine of twice the azimuth of 
ship's heading, measured from the point midway between the 
magnetic meridian and the compass m.eridian. 

The Yariation of the Intensity of Magnetism, for distance 
is known to be Inversely as the Square of the distance. But 
when two magnetic forces act on each other, the intensity of their 
mutual or joint force varies Inversely as the third course of 
their distance. 

Thus — if two magnetic bodies acting on each other at a given 
distance produce a given deviation, then at twice the distance, 
the deviation will be only J as much; at three times the distance, 
it will be only ~ as much; at ten times the distance, ^^ as 
much, etc. 

This law shows us how a small element of disturbance near by, 
can make more trouble than a shipload of iron a little ways off. 

Heeling Deviation. — So far, our consideration of compass 
errors have been considered with regards vessels on an ^^even 
beam. But there is an error resulting from the lifting or heeling 
of the vessel, called Heeling Deviation, 

When the vessel is heeled to starboard or port, the vertical 
longitudinal plane, containing the compass, is shifted to leeward 
of that containing the general center of magnetic effort of the 
ship. As a consequence, the relations existing with ship on an 
even beam are disturbed. 

But, fortunately, our lake vessels have so much more beam for 
their tonnage and draft of water than sea-going vessels, that this 
disturbance will now give us much trouble. 

This error is at its maximum when the ship's center line is 
parallel with the compass needle, as with ship's head IST. or S. 
and nothing with head E. or W. magnetic. 

Mechanical Correction of Deviation, by Means of Mag- 
nets. — Because the two parts B and C, of deviation are semi- 
circular, or nearly so, they may be compensated, or neutralized 
mechanically by introducing magnets acting in contrary direction. 



92 A MANUAI. OF NAVIGATION FOR THE LAKES. 

Having oriented the vessel by any of the methods heretofore 
explained, with ship's head to any one of the cardinal points, — 
say to the North, — mark on the deck under the compass, and on 
the wall of the pilot house, if near, the intersection of the mag- 
netic meridian plane with them, that passes through the compass. 
This line will be parallel to ship's center line. 

Also, at right angles to this line, draw the intersection of the 
transverse vertical plane passing through the centre of compass, — 
marking the trace of same on the side of pilot house, when 
near, — showing a possible place for a magnet. 

Place a magnet with its center on the fore-and-aft line of the 
deck, moving from or towards compass, — changing ends if neces- 
sary, — till the compass point correctly, — and fasten tempor- 
arily, keeping the magnet perpendicular to the meridian plane. 

If the deck is too far off from the compass, for the strength of 
the magnet, apply a larger magnet, or use two magnets, — or 
fasten one to the side or front of the pilot house. 

Then, ship's head being swung to magnetic East ci AVest, 
again bring the needle to its normal place, as befor?, — being 
careful to keep the magnet truly fore-and-aft, with its center on 
the transverse line on the deck, — and fasten temporarily. 

If, now the fixed magnetism of the ship represented by X^ 
and — B in Fig. 1, of Plate II, be in its proper place, with its 
zero points at the North and South, coinciding with the nodal 
points of the other two parts C and D, then compensation for 
the semi-circular deviation will be complete, and ship will 
*' reverse bearings" on the cardinal points, and in the quadrants, 
except for the error due to quadrantal deviation. 

But this is too often not the case. This part B may be shifted 
in azimuth with regard to the other parts C and D, so as to give 
a material error of a J or possibly a whole point on reversal of 
ship. In this case no amount of "fixing" or "fudging" will 
avail us. If Ave bring needle to place on one cardinal point, it 
will be out on the other, — and if we bring it to place on the 
other, it will be out on the one. 

In this case we can do nothing more in the way of correction, 
but resort to a Deviation Card. Swing ship again to verify the 
work, and secure the magnets permanently. 

If the quadrantal deviation (for there is some in all ships), is 
to be corrected by deviation card, which is the better way, swing 



DEVIATION CARD. 



93 



ship again, putting ship's head carefully to all the alternate 
points of the card, magnetic by means of the desired compass, 
and note the reading carefully of the ship's compass, and record 
them. 

If there was no error at the cardinal points, the errors found 
in the quarters, will rarely exceed a J point. But if these courses 
at the cardinal letters, are reversed, these errors will affect the 
•quadrantal errors. 

The following is a deviation card from actual practice, the 
original error being over 2^ p. 

The first and third columns giving the magnetic course desired; 
the second column giving the correction, which must be read 
with the course in the magnetic column, to get the compass 
course desired. 

Thus, to get S. E. magnetic, we take S. E. J S. per compass; 
and for IS". W. magnetic, we take N. W. J N. per compass, etc. 

Steering Card. 



Magnetic 


Correc- 


Magnetic 


Correc- 


Course. 


tion. 


Course. 


tion. 


Compass 


Course. 


Compass Course. 


North. 




South. 




K. N.E. 




s. s. w. 




N. E. 




s. w. 




E. N. E. 




w. s. w. 




East. 




West. 




E. 8. E. 


J S. 


W.N.W. 


JN. 


S. E. 


i s. 


N. W. 


}N. 


S. S. E. 


i s. 


N. N. W. 


JN. 


South. 




North. 





Mechanical Correction of Quadrantal Deviation, is made 
hy means of soft iron, or cast iron. Experience has shown that 
cylinders of cast iron, 3 to 3i inches in diameter, and 
9 to 12 inches lon^, with hemispherical ends, and placed on a 
level with the compass card, and with their ends pointing radially 
to the center of the card, give the best results. 

Nails and chain in boxes have been used; cast iron balls, also, 
have been used with satisfactory results. 



94 A MANUAL OF NAVIGATION FOR THE LAKES. 

The Correction. — The semi-circular deviation having been 
corrected, set ship's head to one of the inter-cardinal points, — 
say N. E. — ^^magnetic. 

Place one of the cylinders to the north of the card, on a 'evel 
with the card, and with end pointing to the center of the 
compass. This corrector should be directly in line of the needle 
when brought to its normal place. 

Place another cylinder to the east or Avest side of the compass, 
in the same manner, as may be necessary to make the compass 
point correctly. 

Now, keeping the ends of the correctors at the same distance 
from the card, — move them both outward or inward till the com- 
pass points correctly, — and the work is done, — secure correctors. 

Theoretically, this correction should be nearly perfect, but 
practically it may be very imperfect, as the result of a non- 
symmetrical arrangement of soft iron in the ship. 

Correction of the Heeling Error. — This is made by means 
of a Vertical Magnet, under the center of the compass. 

Small vessels may be readily heeled 8° to 10°, when a magnet 
may be applied in a line that would be vertical when ship is on 
an even beam, — with that end up that brings the needle to 
place, — and varying the distance, by moving it up or down, till 
the needle shall point correctly, when it may be secured. 

Large vessels cannot be readily heeled. In this case resort 
must be had to a Magnetic Survey of the ship. But the dis- 
cussion of such a survey is beyond our purpose. For this 
information the student is referred to the Admiralty Manual 
for 1874. 

Practical Conclusions. — The following are some of conclu- 
sions drawn from the scientific investigations and long practical 
experience of Messrs. Smith & Evans, of the Liverpool Compass 
Committee. 

(I.) All mechanically corrected compasses, should have com- 
pound needles, — or two parallel needles, whose extremities are 
60° apart. 

(II.) If single needle compasses are to be corrected, the 
needles should not be over six or seven inches in length, 

(III.) A correcting magnet on the same level of the compass, 
should not be nearer to the center of the needle, than six times 
the leno^th of the needle. 



PRACTICAL CONCLUSIONS. 95 

(TV.) In making corrections for quadrantal deviation, the soft 
iron correctors should not be brought nearer to the center of the 
compass, than two times the length of the needle. 

(Y.) No compass that is to be corrected by magnets, should be 
placed where the original deviation is more than two points. 

(A"I.) The compass should not be near either end of an iron 
ship. And if the decks are of iron, they should be provided 
with a hatch immediately below the compass, of a width IJ times 
the height of compass above deck. 

(VII.) If the compass of an iron ship is to be carried amid- 
ships, the direction of the head of ship, during the building, is 
not important. 

Remark, — The preceding is but a meager outline of the sub- 
ject, ''Terrestrial Magnetism," wdth ships, and their compasses, 
which alone would require a large volume, but it is believed to be 
enough to give the shipmaster some idea of the forces that 
disturb his compass, and to put him on his guard in the care of 
that valuable instrument. 

For further information on this topic, he is referred to 
*' Magnetism of Ships" and '' Deviation of Compasses," published 
by the Bureau of Navigation, — Navy Department, 1867. 



CHAPTER VI. 

The Propeller Wheel. 

In view of the great importance of the propeller wheel in pro- 
moting the vast and growing commerce of the lakes, it is desirable 
that a more general, and in some respects, a more correct know- 
ledge of that indispensable agent be had by our maritime people. 
Accordingly I offer the following concerning it. 

Deffinitions. — The Length of the wheel is the distance from 
the forward edge to the after edge of the blade, measured in a 
direction parallel with the shaft, and usually is about ~ of the 
diameter. In the early history of the wheel, it was much longer. 

The Pitch is the distance made, by one revolution of a point, 
with regard to the shaft, and is usually given in terms of the 
wheel, — as a pitch of IJ or 1 J diameters. It is sometimes given in 
feet, but that method gives no idea of the pitch — angle. 

The Net Pitch is the distance made by the vessel, by one 
revolution of the wheel. This is usually given in feet, for the 
purpose of deducing the speed of ship from the number of revo- 
lutions of the wheel, in a given time, though when used for 
comparing the value of different wheels, it should be given in 
terms of the diameter. It is the pitch as usually defined, dimin- 
ished by slip. 

Slip is the difference between the pitch of the wheel, and the 
distance made by the vessel at one turn of the wheel, and is 
expressed in feet, or as a per cent of the pitch, as wanted. 

Example. If a wheel of 9 feet diameter have a pitch of IJ 
diameters, propel a vessel 10 feet with one revolution, it is said to 
have a pitch of 12 feet, — a net pitch of 10 feet, and slip of 2 feet, 
or of 2^12=.166, or say ^. 



CENTER OF EFFORT. 97 

Negative Slip. — Sometimes the vessel shows a speed greater 
than that due to the pitch of her wheel. This apparent 
paradox, — for it is only apparent, — comes from measuring the 
pitch at the end of blade instead of measuring it at a point called 
'the Center of Effort, situated about — to — of the length of 
blade inboard from the end, as will be hereafter explained. 

It is pitiful, yet laughable, to see the absurd theories that have 
been advanced, even by learned professors, to account for this 
apparent anomaly. 

True Screw. — In this screw, the working face is a warped 
surface, generated by a line having two motions, — one angular at 
a uniform rate, and the other, a motion of translation at a uniform 
rate, along a right line, and always making the same angle with 
that line. 

Illustration. — In Fig. 31, let m. n. represent the center line of 
shaft, or hub of wheel, and let the line n. a. A. represent the 
generating line. 

Then if this line be moved uniformly along the line n. m., and 
with a uniform angular motion along A. C. as indicated by the 
arrow points, it will generate the helicoidal surface, A. c. m. n. 
forming one surface of the blade of a true screw, — A. C. being 
the Helix. 

Pitch Angle.— In Fig. 31, let A. B. be the arc of a circle 
whose plane is at right angles to the line m. n. or shaft, — and the 
arc A. C. the helix generated by the point A. in its revolution 
about the line m. n. — then is B. A. C. the pitch angle, and 
B. C, parallel with and equal to n. m., is the pitch corresponding 
to Avidth of blade. 

Center of Effort is that point in the length of the blade, 
having the same amount of work on either side of it, and, in the 
true screw, is at — of length of blade from center of wheel, — and 
not at — as told us in engineering books. 

If the wheel were quiessent, supporting a weight uniformly 
distributed over the disk area of the same, — then the center of 
effort would be at — of radius from center, as said. But the cir- 
cular motion throws this center outboard. 

The disc area of the wheel varies as the square of the diameter. 
Also the intensity of percussion varies as the square of velocity 
of the several points, in the radius of the wheel. And the 
amount of work done varies with the product of these two 



98 A MANUAL OF NAVIGATION FOR THE LAKES. 

factors, — whence the total work up to any point in the radius of 
the wheel, varies as the fourth power of the distance of such 
point from the center of wheel. 

As a consequence, the center of effort is at a point equal to the 
fourth root of J := — of R., tliat is, in the helix a. b. — of rad- 
ius from n. m. 

The angle which this helix a. b. makes with the plane of the 
arc A. B. and which is at right angles to the shaft, is the angle 
that should always be used in any calculations concerning the 
efficiency of wheels, or the amount of power wanted for 
propulsion. 

AVhen the pitch is measured at that point, we shall never hear 
of negative slip. 

Expansion of Pitch, Radial and Axial, is a change of the 
pitch from that of the true screw. 

If the blade (Fig. 31) be made narrower at the inboard end, by 
lifting the point n. towards m. the pitch of the blade at that end 
will be reduced. 

This change is called Radial Expansion, and is introduced by 
many wheel makers, for the purpose of throwing the work of this 
end of blade out to where the pitch-angle is smaller, to avoid 
part of the loss from oblique action. 

But there is a limit to this reduction, it must not exceed the 
prospective slip. The object is to relieve the inboard end entirely 
of work, which is attained by making the pitch the same as the 
net pitch. If this is made less than the net pitch, the blade takes 
water on the forward face, — doing negative work. 

Axial Expansion, is an increment of pitch from the forward 
edge to after edge of blade, making the working face concave. 

The object is to take hold of the water gradually, and to in- 
crease the pressure by increasing the pitch gradually, in conse- 
quence of which the helix AC. (Fig. 31) is concave towards AB. 

This expansion is all right in the hands of those who know 
how to use it, for it has its limit, which is the prospective slip. 
If the expansion exceed this limit the blade takes water on the 
forward face, doing negative work. Besides, the propelling of 
ship is done with the after part of blade, where the pitch angle is 
greater, — and where, consequently loss from oblique action is 
greater. 



KAPIAIi EXPANSION. 



99 



This is probably the most disasterous leak the coal-bunkers of 
the lakes have ever been called on to pay. This mistake has 
been general. 

Fortunately, our best wheel makers know better now than to 
exceed this limit, — while others, perhaps afraid of it, do not give 
their blades any axial expansion at all, — which is perhaps as 
well, for the smaller the slip, the more nearly must the blade be 
a true screw. 



■yvV 




Fig. 31. 

By means of radial expansion, the center of effort is moved 
outward from its normal place in the true screw, but just how 
much, owing to the different ways of doing it, and the different 
degrees to which it is done, it would be difficult to say. The prob- 
ability is we may locate the center of effort between — and ^^ of 
the distance from center to end of blade. 



100 A MANUAL OF NAVIGATION FOR THE LAKES. 

To Measure the Pitch, is frequently required for the pur- 
pose of comparing different wheels. It is of importance for 
every engineer, master and owner of a steamer, to know the 
pitch of his wheel, so as to know, in the event of a break, how to 
order a new one. 

This involves a knowledge of the angle BAG. (Fig. 31.) or of 
the sides of that triangle. 

The angle may be found by applying a carpenter's bevel, so as 
to have one arm of the bevel on the helix AC. and the other on 
the plane AB. and both at right angles to the radial at the point 
of application, as at A. 

The angle being found, construct on paper, as BAG. From 
any point B. set up the perpendicular BC. Then is BC. the 
pitch due to the developed helix BG. 

Divide the base AB. by 3.14, then we have the diameter of a 
circle whose circumference is AB. Divide BG. by the quotient 
thus found, when we have the pitch in terms of the diameter of 
the wheel. Example. 

Suppose we find AB. in (Fig. 32.) by scale measure to be 25J 
inches, and BC. to be 12 inches, then 

25J^3.14=8 and 

12 -^ 8 =1J. That is, our wheel has a 

pitch of IJ diameters. It is the pitch at the end of blade, and is 
the pitch by which wheels are usually defined. But wheels must 
be compared by means of the angle at the center of effort, called 

The Working' Pitch Angle, which is always greater than 
that found at end of blade, by — to —^ depending on the amount 
of radial expansion. 

This angle is readily found in degrees, — thus 

Divide the ratio of pitch to diameter, as 1, IJ, IJ, etc., by 
3.1416, and we have the tangent of pitch angle at end of blade. 

Increase this tangent by — or — , as may be required and we 
have tangent of the Working Pitch Angle. 

Example for a wheel whose pitch is IJ diameters: 
1.5--3.14163=.4774=Tan. 25.° 31^ 
which is angle at end of blade. 

Increasing this tangent by its — , we have 

.4774X'-^^^--=.o374=:=Tan. 28.° 08^ 
for angle at c Alter of effort. 

Or measure by protractor, or scale of chords. 



HOLDING POWER. 101 

The "HoldJug* Power," or the ability of the wheel to furnish 
inertia for the engine to work upon, depends on the disc area of 
the wheel, and the depth of water draAvn by it. vSo that when 
the wheel runs ''awash," the Holding Power varies as the 
cube of the diameter, or 

If the draft of the wheel is greater than the diameter, then the 
Holding Power will vary as the square of the diameter, mul- 
tiplied by the draft. 

Illustration. — If a wheel of given diameter give a certain 
amount of inertia, running *' awash/' then a wheel twice as large 
under the same condition, will give eig'llt times as much; and a 
unit of area in the large wheel, as a circular foot of the disc area, 
wdll have twice as much holding power as the same unit in the 
small wheel, for it has twice the weight of water on it. 

The Holding Power per I. H. P., furnished by different wheel 
makers, is widely different in the amount, — varying from 1 to 5 
cylindric feet. 

The minimum furnished by the best designers at present, is the 
inertia to be derived from about 1 cylindric foot of water per 
I. H. P., though the average of wheels on the lakes, is consider- 
able above that. 

The wheels of the steamers ''Alaska,'" "Spree," and the U. S. 
Steamship "Maine," are so proportioned as to give about 1 
cylindiic foot per I. H. P., — whence, to determine the diam- 
eter of the wheel, in feet for that amount of inertia, or holding 
power, we have only to take the cube root of the !• H. P. 

Or, if the wheel be deeply submerged, we may divide the 
I. H. P. by the draft, and take the square root of the quotient. 
Example: 

The wheels of the U. S. Steamship "Maine," are calculated to 
work off 9,000 Horse Power, or 4,500 each. And the vessel is to 
draw 21 J feet of water. Dividing the 4,500 by the draft, which 
is say 20 feet, we have 225, as the square of the diameter, — the 
square root of which is 15 feet, as provided by her designers. 

Difficulty in ProYiding" Holding- Power for small vessels. 
AVe know from Geometry, that similar solids or volumes vary as 
the third power of any of their similar dimensions. Also, that 
surface of similar figures are as the second power of their similar 
sides. 



102 A MANUAL, OF NAVIGATION FOR THE LAKES. 

And we know that when two vessels are similar, and loaded to 
the same per cent, of their depth of hold, that their displacement 
volumes are similar, and therefore that their wet surfaces are 
similar, and have the relation of the squares of any of their sim- 
ilar dimensions. Whence, the power required for them will be 
as the squares of those dimensions, while as we have seen, the 
^^ holding power'' varies as the cube of those dimensions. Hence 
we see that the holding power changes twice as fast as the 
requirements for propelling power. Illustration. 

Say a vessel with 8 feet draft, is provided with a satisfactory 
power, and wheel, 

A vessel of 16 feet draft, would have four times as much 
power, if provided in proportion to increase of surface, while 
the inertia available, is eig'llt times as much, i. e. the inertia 
available, varies twice as fast as the power required. 

This law is not generally known, and what is worse, it cannot 
be repealed, nor circumvented. 

If it w^ere better understood, people would know why a ton of 
displacement in a small vessel, requires so much more power for 
a given speed, than it does in a large one, — which is simply be- 
cause the inertia required, is not obtainable. 

With this rate of change, between the power required, and the 
inertia available, there is some point, of course, where the inertia 
to be had, Avill meet the requirements, when the wheel is running 
^' awash." 

This condition will be found with vessels drawing 7 to 8 feet of 
water, i. e. with such vessels provided say wdth 7 feet wheels, all 
the inertia required, is available, but not more. 

With vessels drawing more water, the supply increases faster 
than the demand, — i. e. as the cube of the draft, while the 
amount wanted is only as the square of the draft. 

AVith vessels of smaller draft, the supply falls short of the 
requirements, by the same ratio. Example: 

With a vessel of only 4 feet draft, the power required, to be in 
the proportion of their surfaces if the vessels were similar, would 
be as the square of J, which is J, while the inertia available 
would the cube of J=J, that is, the supply falls short of the 
demand, in this case, in the ratio of J to J, or 1 to 2, i. e. we can 
only get half of what we want. 



USEFUL AND LOST WORK. 



103 



Many plans have been devised to surmount this difficulty, — 
chief among which is the increase of the pitch. This introduces 
two other troubles, loss by oblique action of blade, and a serious 
lateral pull, by the great amount of lost work in the lower blade, 
resulting in loss by great slip. 




Fig. 32. 

Two wheels have been introduced, but this is very expensive, 
besides it increases the exposure to injury. The expense is seen 
when we multiply the diameter of the one wheel, say 8 feet, by 
the cube root of J=.794, and find that we must use two wheels, 
6 J feet diameter, to get the holding power afforded by one 8 feet 
diameter, when the wheels run ^^ awash." 



Useful and Lost Work. — To represent these parts, let the 
line DC. in Fig. 32, be the helix at center of effort. 



104 A MANUAL. OF NAVIGATION FOR THE LAKES. 

From any point in this line, and at right angles to it, draw the 
line c. d. to represent the total work of blade. Divide into 100 
equal parts. Through d. draw d. b. parallel with BC, and 
through c. draw b. c. parallel with D. B. 

Then will b. d. represent the amount and direction of the use- 
ful work, and b. c. those of the lost work. 

The total, or 100 parts of work must be divided on these lines, 
in proportion to their squares. This is done by demitting a per- 
pendicular from b. onto d. c, showing in this case, the useful 
work to be about — of the total work. The lost work, repre- 
sented by b. c. is expended entirely in lateral work, like a paddle 
wheel turned fore-and-aft and which is very much increased by 
increasing the pitch angle, is seen to be — , in this example. 

The triangles BCD. and b. c. d. being alike, the side b. c. cor- 
responding to pitch of blade, is the Sine of the Pilch Angle, 
while d. c. is cosine, whence from the preceding, we have 

Cosine^ of the working pitch angle, measures the useful work, 
and siue^ measures the lost work, in per cent of the whole. 
In this manner was the following table made: 



Pitch in 
Diameter. 


Angle. 


Per Cent. Loss. 


1 


19° 


.11 


li 


24° 


.16 


H 


28° 


.22 


1} 


32° 


.28 


2 


36° 


.35 



Thus, it is seen from the table that we more than treble the 
loss from oblique action of blade, by doubling the pitch. 

Advantage of a Large TVlieeL — It is not infrequently desir- 
able to make the wheel larger in diameter than the preceding 
considerations would indicate, not necessarily for the purpose of 
getting more inertia for the engine to work upon, — but to keep 
the pitch-angle within economic limits. 

When it is desired to obtain great speed, we must have great 
pitch in some form. We may either increase the piston -speed, 
by which more revolutions of the wheel are attained in a given 
time, or we increase the diameter of the wheel, — keeping the 



ADVANTAGE OF A LARGE WHEEL. 105 

pitch-angle constant, — till the desired pitch is attained. Another 
method, practiced by some, — who have yet their first lesson tO' 
learn concerning the propeller wheel, — is to increase the pitch- 
angle. But as this increases the loss from oblique action of 
blade very fast, after a pitch of about IJ diameters has been 
attained, it is not to be tolerated, above that limit. 

It is true that increase of diameter is objectionable on some 
accounts, — as increased first cost, — increased exposure to injury, — 
increase of friction; — but reduction of loss by oblique action of 
blade, i. e. a saving of power, gained thereby; is greater than all. 



Explanation of the Tables. 



I^^ Is a traverse table, giving the differences of latitude and 
departure for distances up to 10, and bv courses varying by 4 
point up to 8 points. For illustration of their use, see pages 
41 to 43. 

II. Is a table of the natural functions, — Sine and Cosine,— 
Tangent and Cotangent, — Secant and Cosecant, for every 5 of the 
quadrant, and to four places of decimals, — more decimals, or 
shorter intervals of arc, for purposes of azimuth, with our lake 
compasses being entirely unnecessary. 

For further explanations of their use, see pages 11 to 13. 

III. Is a table of rhumbs, varying by J point, for converting 
points to degrees, and the reverse; — with their Sines and Cosines, 
Tangents and Secants, — useful in making computations for angles 
in points, etc., instead of in degrees, — as in Table II. 

IV. Is a table of amplitudes of the sun, for the latitude of the 
lakes, with the time of his rising and setting. Also the rate of 
the change of his azimuth from sunrise, to the time of his cross- 
ing the prime vertical, when the latitude and declination are 
alike. 

This rate being practically uniform for all declinations in any 
latitude above about 40°, affords the means for deducing a table 
of time azimuth, good for one to three hours, from a single 
amplitude; — thereby vastly enlarging the utihty of amplitudes. 

For further explanations see pages 71 to 73. 



EXPLANATION OF THE TABLES. 107 

V. Is a table of the sun's declination to the nearest minute, 
for the current year (1891), with the corresponding equation of 
time, — the sign prefixed, being that for reducing mean time to 
apparent time; — wanted with the table of time azimuth, when 
orienting ship. 

This table is never precisely correct, except for the particular 
date for which it was computed; but as it is the mean between 
two leap-years, it can never be far wrong, — the change of declin- 
ation for 12 hours, being the maximum error that can occur 
during any four years, — after which interval, the table is very 
nearly correct again. 

Ignoring this small error, — as we may for all purposes of azi- 
muth, this table will be available for a great many years. 

VI. Is a table of time azimuths of the sun, for the latitude of 
the lakes. Useful in orienting ship for compass errors. — See 
pages 74, 77, 78. 

YII. Is a table of time corrections for reducing standard time 
to local mean time. It is computed by reducing the difference of 
longitude between any locality, and the longitude of the standard 
time, to time, and applying the sign that reduces the standard 
time to local mean time. Thus: 

The difference of longitude between the light-house at Chicago, 
and the longitude of central standard time, is 90^ — 87°, 37^=2°, 
2o^ , or 9^^, 32^ of time. Then because the standard meridian is 
^vest of Chicago, this difference must be applied with the plus 
sign, i. e. we must add 9^^, 32^ to central standard time, to obtain 
the mean local time for Chicago. In this manner was the table 
constructed. It is wanted with the equation of time in using the 
tables of time azimuths. See page 74. 

VIII. Is a table of chords for setting off, or measuring angles. 
For particulars see pages 12-13. 

IX. Is a table of meridianal parts, for the construction of 
Mercator's chart. It is given for every 2 minutes of latitude 
from to 75°. The intermediate minute is readily found by 
interpolation, Avhen wanted. See pages 48 to 52. 

X. This table gives the value of 1 minute of longitude for 
€ach degree of latitude up to 30°, and for each half degree 
thence up to 75°. It is wanted in many questions of dead reckon- 
ing for reducing departure to difference of longitude. See 
pages 44 to 46. 



108 A MANUAL OF NAVIGATION FOR THE LAKES. 

XI. Is a table for the correction of middle latitude, in middle 
latitude sailing. See pages 46-47. 

XII. Is a table for finding distance of objects from ship, from 
two bearings, and distance sailed between them. Pages 59 to 61. 

XIII. Is a table for reducing difference of longitude to time, 
wanted in deducing local apparent time of a locality from the 
longitude of the locality and standard time. 

Tariatioil Cliartlet. — Wishing to amplify the subject of vari- 
ation somewhat, I give the following as an explanation of the 
chartlet of ''Magnetic Tariations." 

First, to explain the terms right and left as applied to the 
magnetic card. When the north end of the needle is to the right 
of the true north, it is in east variation, and in west variation 
when the north end is to the left of the true north. This is well 
when the ship has northings in her course, but it is a more gen- 
eral expression to say that the card is out to the right or left, as 
the case may be, by variation. 

It Avill be observed that in sailing either wp or down the lakes 
we are continually crossing the isogonic curves. In going west 
the card swings to the right 18J° between the foot of Lake 
Ontario and the head of Lake Superior, — thereby leading the 
vessel to the right or north. 

In the same manner, in going from Duluth to Buffalo, the card 
swings to the left, taking the vessel to the left, or north, as 
before. 

That is to say, — in going any way across these lines, we are 
taken to the north by change of the variation. 

This *^ change" of variation, is a very different thing from 
variation itself. When we go west, with a westerly variation, we 
are led to the Sonth, and to the north with an easterly varia- 
tion, — unless correction is made, — and conversely. 

But wdth the *' change," we are led to the north in any case. 
And the track made by the ship as the result of this *^ change," 
is a curve, like a railroad curve, precisely. And the straight 
line joining the end of this curve, changes its direction, with a 
change in the length of the curve, just half as fast, as the curve 
changes its direction. Whence, 

Correction is made for this *' change," by taking half the sum 
of the variation at the two extremities of a run for the mean 
variation to be applied at either end of the run. 



VARIATION CHARTLET. 109 

Thus:— Toledo to Buffalo,— Variation at Toledo,=rO°. Varia- 
tion at Buffalo, :=4J° W. The J sum=2J° W. is the mean 
variation to be applied in going either way between these ports. 

In ''Shaping Course," the following rule w411 always apply 
for mean variation. 

With variation to the right, apply correction to the left. 
With variation to the left, apply correction to the right. 

There is scarcely another place in the world where the 
'^change" of variation must be taken into account as an indepen- 
dent factor. This results from the fact that there is no other 
place in the world where the isogonic curves are so close together, 
or are crossed so directly by vessels, as on our lakes. 



1 

H 

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«(NiH©?0?^tHOCO(Mt-IOMC^tH© 


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3 


3.99 
3.98 
3.95 
3.92 
3.88 
3.83 
3.77 
3.G9 
3.G2 
3.53 
3.43 
3 . 33 
3.21 
3.08 
2.9G 
2.83 


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098 
19G 
294 
390 
48() 
580 
G74 
7()5 
855 
943 
.03 
.11 
.19 
.27 
.34 
.41 


3 


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ClCiXXt-iC'*CMOXi0C0Ot-'*O 

CiCiC5C5CiC;CiCiC^XXXXt-t-t- 


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tHC^COOiHC^COCtHCOMCi-IC^COC 
rH C^ CO -si 




^1 





T.A.BIjE II J^, 




Natural Sines anc 


Cosines. 


^ 


0° 


1° 


2° 


3° 


4° 


' 


Sine, 


Cosine 


Sine 


Cosine 


Sine 


Cosiae 


Sine 


Cosine 


Sine 


Cosine 


Oj .0000 


Unit. 


.0174 


.9998 


.0349 


.9994 


.0523 


.9986 


.0698 


.9976 


60 


5^ 


.0014 


Unit. 


.0189 


.9998 


.0363 


.9993 


.0538 


.9985 


.0712 


.9975 


55 


10 


.0029 


Unit. 


.0120 


.9098 


.0378 


.9993 


.0552 


.9985 


0727 


.9974 


50 


15 


.0044 


.9999 


.0218 


.9998 


.0393 


.9992 


.0567 


.6984 


.0741 


.9973 


45 


20 


.0058 


.9999 


0233 


.9997 


.0407 


.9992 


.0581 


.9983 


.0756 


.9971 


40 


25 


.0073 


.9999 


.0247 


.9997 


.0422 


.9991 


0596 


9982 


.0770 


.9970 


35 


30 


.0087 


.9999 


.0262 


.9996 


.0436 


.9990 


.0610 


.9985 


.0785 


.9969 


30 


35 


.0102 


.9999 


.0276 


.9996 


.0451 


.9989 


.0625 


.9980 


.0799 


.9968 


25 


40 


.0116 


.9999 


.0291 


.9996 


.0465 


.9989 


.0639 


.9979 


.0814 


9968 


20 


45 


.0131 


.9999 


.0305 


.9995 


.0480 


.9988 


.0654 


.9978 


.0828 


.9966 


15 


50 


.0145 


.9998 


.0399 


.9995 


.0494 


.9988 


.0668 


.9977 


.0843 


.9964 


10 


55 


.0160 


.9998 


.0334 


.9994 


.0509 


.9987 


.0683 


.9976 


.0857 


.9963 


5 


60 


.0174 


.9998 


.0349 


.9994 


.0523 


.9986 


.0697 


.9975 


.0872 


.9962 





> 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


bine 


Cosine 


Sine 


Cosine 


Sine 


' 


89° 


88° 


87° i 


86° 


85° 


~b 


5" 


6° 


7° 


8° 


1 9° 


' 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


.0872 


.9962 


.1045 


.9945 


.1219 


.9925 


. 1392 


.9903 


.1564 


.9877 


60 


5 


.08.^6 


.9961 


.1060 


.9944 


.1233 


.9924 


.1406 


.9901 


..1579 


.9875 


55 


iq 


.0900 


.9959 


.1074 


.9942 


.1248 


.9922 


.1420 


.9899 


.1.593 


.9872 


50 


15 


.0915 


.99.=^8 


.1089 


.9941 


.1262 


.99_>0 


.1435 


.9896 


.1607 


.9870 


45 


20 


.0929 


.9957 


.1103 


.9939 


.1276 


.9918 


.1449 


.9894 


.1622 


.9868 


40 


25 


.0944 


.9055 


.1118 


.9937 


.1291 


.9916 


.1464 


.9892 


.1636 


.9865 


35 


30 


.0958 


.99.54 


.1132 


.9936 


.1305 


.9914 


.1478 


.9890 


.16.50 


.9863 


30 


35 


.0973 


.99.53 


.1146 


.9934 


.1320 


.9912 


.1492 


.9888 


.1665 


.9860 


25 


40 


.0987 


.9951 


.1161 


.9932 


.1334 


.9911 


.1.507 


.9886 


.1679 


.9858 


20 


45 


.1002 


.99.50 


.1175 


.9931 


.1348 


.9909 


.1521 


.9884 


.1693 


.9856 


15 


50 


.1016 


.9948 


.1190 


.9929 


.1363 


.9907 


.1535 


.9881 


.1708 


.9853 


10 


55 


.1031 


.9947 


.1204 


.9927 


.1377 


.9005 


. 1550 


.9879 


.1722 


.98.51 


5 


69 


.1045 


. 9945 


.1219 


.9925 


.1392 


.99 3 


.1564 


9877 


.1765 


9848 





Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


84° 


83° 


82° 


8P 


80° 


^ 


10- 


ii° 


l^o 


13- 


14° 


' 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 





.1736 


.9848 


.1908 


.9816 


..2079 


.97H1 


.2249 


.9744 


.2419 


.9703 


60 


5 


.1751 


.9845 


.1922 


.9813 


.2093 


.9778 


.2264 


.9740 


.2433 


.9699 


55 


10 


.1765 


.9843 


..1937 


.9811 


.2108 


.9775 


..2278 


.9737 


.2447 


.9696 


50 


15 


.1779 


.9S40 


.1951 


.9808 


.2122 


.9772 


.2292 


.9734 


.2461 


.9692 


45 


20 


.1794 


.8438 


.1965 


.9805 


.2136 


.9769 


.2306 


.9730 


.2476 


.9689 


40 


25 


.1808 


.9835 


.1979 


.9802 


.2150 


.9766 


.2320 


.9727 


.2490 


.9685 


85 


3^ 


.1822 


.9832 


.1994 


.9799 


.2164 


.9763 


.2334 


.9724 


.2504 


.9681 


30 


35 


.1837 


.9830 


.2008 


.9696 


.2179 


.9760 


.2349 


.9720 


.2518 


.9678 


25 


40 


.1851 


.9827 


.2022 


.9793 


.2193 


.9757 


.2363 


.9717 


.253 J 


.9674 


20 


45 


.1865 


.9824 


.2036 


.9790 


.2207 


.9753 


.2377 


.9713 


.2546 


.9670 


15 


5^ 


.1879 


.9822 


.2051 


.9787 


.2221 


.9750 


.2.391 


.9710 


.2560 


.9667 


10 


55 


.1894 


.9819 


.2065 


.9784 


.2235 


.9747 


.2405 


.9706 


.2.574 


.9663 


5 


60 


1908 


.9816 


.2079 


.9781 


.2249 


.9744 


.2419 


.9707 


2588 


.9659 





^ 


Cosine 


Sine 


Cosine 


Sine 


Cosine Sine 


Cosine 


Sine 


Cosine 


Sine 


' 


79° 


78° 


T7^ 


76" 


75° 



Natural Sines and Cosines — Continued. 


^ 


10" 


16° 


17- 


18° 


19° 


60 
55 
50 
45 
40 
35 
30 
25 
20 
15 
10 




Sine 


Cosine 


Sine 


Cosine 


Sine 


CoMne 


Sine 


Cosine 


bine 


Cosi' e 



5 
10 
15 
20 
25 
30 
35 
40 
45 
50 
55 
60 


.2588 
.2602 
.2616 
.2630 
.2644 
.2658 
.2672 
.2686 
.2700 
.2714 
.2728 
.2742 
.2756 


.9659 
.9655 
.9652 
.9648 
.9644 
.9640 
.9636 
.9632 
.9628 
.9625 
.9621 
.9617 
.9613 


.2756 

.2770 
.2784 
.2798 
.2812 
.2826 
.2840 
.2854 
.2868 
.2882 
.2896 
.2910 
.2924 


.9613 
.9608 
.9605 

smo 

.9.=)96 
.9592 
.9.588 
.9.584 
.9580 
.9576 
.9571 
.9567 
.9563 


.2924 
.2938 
.29.51 
..2965 
.2979 
.2993 
.3007 
.3021 
.3035 
.2049 
.3062 
.3076 
.3090 


.9.563 
.9559 
.9554 
.9550 
.9546 
.9541 
.9537 
.9533 
.9528 
.9.524 
.9519 
.9.515 
.9511 


..3090 
.3104 
.3118 
.3132 
.3145 
.3159 
.3137 
.3187 
.3201 
.3214 
.3228 
.3242 
.3255 


.9511 
.9.506 
.9.501 
.9497 
.9492 
.9488 
.9483 
.9479 
.9474 
.9469 
.9465 
.9460 
.9455 


.3256 
.3269 
.3283 
.3297 
.3311 
.3324 
.3338 
.3352 
.3365 
.3379 
.3393 
.3406 
.3420 


.9455 
.94.50 
.9446 
.9441 
.9436 
.9431 
.9426 
.9421 
.9417 
.9412 
.9407 
.9402 
.9397 


\ 


Cosine 


Sine 


Cosine 


* Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosinej bine 


/ 


74° 


73° 


72° 


71° 


70° 


1 

10 
15 
20 
25 
30 
35 

i 

55 

60 


2U° 


21° 


22° 


23° ) 24° 


/ 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


blue 


Cosint 


.3420 
.3433 
.3447 
.3461 
.3475 
.3488 
.3502 
.3516 
.3529 
.3543 
.3556 
.3570 
.3584 


.9397 
.9392 
.9387 
.9382 
.9377 
.9372 
.9367 
.9362 
.9356 
.9351 
.9346 
.9341 
.9336 


.3584 
.3597 
.3611 
.3624 
.3638 
.3652 
.3665 
.3678 
.3692 
.3706 
.3719 
.3733 
.3746 


.9336 
.9331 
.9325 
.9320 
.9315 
.9309 
.9304 
.9299 
.9293 
.9288 
.9283 
.9277 
.9272 


.3746 
.3759 
.3773 
.3786 
.3800 
.3813 
.3827 
.3840 
.3854 
.3867 
.3881 
.3891 
.3907 


.9272 
.9266 
.9261 
.9255 
.9250 
.9244 
.9239 
.9233 
.9228 
.9222 
.9216 
.9211 
.9205 


.3907 
.3921 
.3934 
.3947 
.3961 
.3974 
.3987 
.4001 
.4014 
.4027 
.4041 
.4054 
.4067 


.9005 
.9199 
.9194 
.9188 
.9182 
.9176 
.9171 
.9165 
.9159 
.9153 
.9147 
.9141 
9135 


: .4067 
.4081 
.4094 
.4107 
.4120 
.4134 
.4147 
.4160 
.4173 
.4187 
.4200 
.4213 
.4226 


.9135 
.9127 
.9124 
.9118 
.9111 
.9106 
..9100 
.9094 
.9087 
.9081 
.9075 
.9069 
.9063 


60 
55 
50 
45 
40 
35 
30 
25 
20 
15 
10 
5 



vJosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine] Sine 


Cosine 


Sine 


f 


69° 


68° 


67° 


66° 


65° 


\ 

~0 
5 

10 
15 
20 
25 
30 
35 
40 
45 
50 
55 
60 


25° 


26° 


27° 


28° 


29° 


f 

60 
55 
50 
45 
40 
35 
30 
25 
20 
15 
10 
5 



Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


.4226 
.4239 
.4252 
.4266 
.4279 
.4292 
.4305 
.4318 
.4331 
.4344 
.4357 
.4371 
.4384 


.9063 
.9057 
.9051 
.9045 
.9038 
.9032 
.9026 
.9020 
.9013 
.9007 
.9001 
.8994 
.8988 


.4384 
.4397 
.4410 
.4423 
.4436 
.4449 
.4462 
.4475 
.4488 
.4501 
.4514 
.4227 
.4540 


.8988 
.8982 
.8975 
.8969 
.8962 
.8956 
.8949 
.8943 
.8936 
.8930 
.8923 
.8917 
.8910 


.4540 
.4553 
.4.566 
.4579 
.4592 
.4605 
.4617 
.4630 
.4643 
.4656 
.4669 
.4682 
.4695 


.8910 
.8903 
.8897 
.8890 
.8883 
.8877 
.8870 
.8863 
.8857 
.8850 
.8843 
.8836 
.8829 


.4695 
.4708 
.4720 
.4733 
.4746 
.4759 
.4772 
.4784 
.4797 
.4810 
.4823 
.4835 
.4848 


.8829 
.8823 
.8816 
.88u9 
.8802 
.8795 
.8788 
.8781 
.8774 
.8767 
.8760 
.8753 
.8746 


.4848 
.4861 
.4873 
.4886 
.4899 
.4912 
.4924 
.4936 
.4949 
.4962 
.4975 
.4987 
.5000 


.8746 
.8739 
.8732 
.8725 
.8718 
.8711 
.8704 
.8696 
.8689 
.8682 
.8675 
.8667 
.8660 


Cosine 


Sine 


Cosme 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


64° 1 


63° 


62° 1 61° 


60° 





T^BXjE II J^, 


Natural Sines and Cosines — Continued. 


"o 


30" 


31° 


32^ 


33^ 


34° 


60 


Sine 
.5000 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


v^osine 


.8660 


.5150 


.8572 


.5299 


.8480 


.5446 


.8387 


.5592 


.8290 


5 


,5013 


.86.53 


.5163 


.8564 


.5311 


.8473 


.5459 


.8379 


.5604 


.8282 


55 


10 


.5025 


.8646 


.5175 


.8557 


.5324 


.8465 


.5471 


.8371 


.5616 


.8274 


10 


15 


.5038 


.8638 


.5188 


.8549 


.5336 


.8457 


.5483 


.8363 


.5628 


.8266 


45 


20 


.6050 


.8631 


.5200 


.8542 


.5348 


.8449 


.5495 


.83.55 


..5640 


.8258 


40 


25 


.5063 


8624 


.5213 


.8534 


.5361 


.8442 


..5507 


.8347 


.5652 


.8249 


35 


30 


.5075 


.8616 


.5225 


.8526 


..5373 


.8434 


..5519 


.8339 


.5664 


.8241 


30 


35 


.5088 


.8609 


.5237 


.8519 


.5386 


.8426 


.5531 


8331 


.5676 


.3233 


25 


40 


.5100 


.8601 


.5250 


.8511 


.5397 


.8418 


.5544 


.8323 


.5688 


.8225 


20 


45 


.5113 


.8594 


.6262 


.8503 


.5410 


.8410 


.55.56 


.8315 


.5700 


.8216 


15 


50 


.5125 


.8587 


.5274 


.8496 


.5422 


.8402 


.5.568 


.8307 


.5712 


.8208 


10 


65 


.5138 


.8579 


.5287 


.8488 


.5434 


.8395 


.5580 


.8298 


.5724 


.8200 


5 


60 


.5150 


8572 


.5299 


.8480 


.5446 


.8387 


.5592 


.8290 


.5736 


.8191 


_0 

/ 


^ 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


59° 


58° ! 5' 


JO 


56° 


55° 


^ 


35° 


36° 


37° 


38° 


39° 


~^ 


Sine 


Cosine 


Sine 


Cos ne 


Sine 


Cosine 


Sine 


Cosine 


Sine Cosine 





.5736 


.8191 


.5878 


.8090 


.6018 


.7986 


.6157 


.7880 


.6293 


.7771 


60 


5 


.5747 


.8183 


.5890 


.8082 


.6030 


.7978 


.6168 


.7871 


.6304 


.7762 


55 


10 


.5760 


.8175 


.5901 


.8073 


.6041 


.7969 .6179 


.7862 


.6316 


.7753 


50 


15 


.5771 


.8166 


,5913 


.8064 


.6053 


.7960 1 .6191 


.7853 


.6327 


.7744 


45 


20 


.5783 


.8158 


.5925 


.8056 


.6064 


.7951 


.6202 


.7844 


.6338 


.7735 


40 


25 


.5795 


.8158 


.5936 


.8047 


.6076 


.7942 


.6214 


.7835 


.6350 


.7726 


35 


30 


.5807 


.8141 


.5948 


.8039 


.6088 


.7933 


.6225 


.7826 


.6361 


.7716 


30 


35 


.5819 


.8133 


.5960 


.8030 


.6099 


.7925 


.6236 


.7817 


.6372 


.7707 


25 


40 


.5831 


8124 


.5972 


.8021 


.6111 


.7916 


.6248 


7808 


.6383 


.7698 


20 


45 


.5842 


.8116 


.5983 


.8012 


.6122 


.7905 


.6259 


.7799 


.6394 


.7688 


15 


50 


5854 


.8107 


.5995 


.8004 


.6134 


.7898 


.6271 


.7790 


.6406 


.7679 


10 


55 


.5866 


.8099 


.6006 


.7995 


.6145 


.7889 


.6282 


.7781 


.6417 


.4670 


5 


60 


.5878 


.8090 


.6018 


.7986 


.6157 


.7880 


.6293 


.7771 


.6428 


.7660 
Sine 


_0 

/ 


^ 


Cosine 


Sine 


v^Oaine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


54° 1 


53° 


52° 1 


5] 


° 


50° 


~0 


40° 


41° 


42° 1 


43° 1 


44° 


/ 


Sine 
.6128 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


.7660 


.6561 


.7547 


.6691 


.7431 


.6820 


.7313 


.6947 


.7193 


60 


5 


.6439 


.7651 


.6572 


.7537 


.6702 


.7422 


.6831 


.7304 


.6957 


.5183 


55 


10 


.6450 


.7642 


.6582 


.7528 


.6713 


.7412 


.6841 


.7294 


.6967 


.7173 


50 


15 


.6461 


.7632 


.6593 


.7518 


.6724 


.7402 


.6852 


.7284 


.6978 


.7163 


45 


20 


.6472 


.7623 


.6604 


.7.509 


.6734 


.7392 


.6862 


.7274 


.6988 


.7153 


40 


25 


.6483 


.7613 


.6615 


.7499 


.6745 


.7383 


.6873 


.7364 


.6999 


.7143 


35 


30 


.6494 


.7604 


.6626 


.7490 


.6756 


.7363 


.6883 


.7254 


.7009 


.7132 


30 


35 


.6505 


.7595 


.6637 


.7480 


.6766 


.7373 


.6894 


.7244 


.7019 


.7122 


25 


40 


.6.517 


.7585 


.6648 


.7470 


.6777 


.7353 


.6905 


.7234 


7030 


.7112 


20 


45 


.6528 


.7576 


.6659 


.7461 


.6788 


.7343 


.6915 


.7224 


.7040 


.7092 


15 


50 


.6.539 


.7566 


.6690 


.7451 


.6790 


.7333 


.6926 


.7214 


.7050 


.7092 


10 


55 


.6550 


.7557 


.6680 


.7441 


.6809 


.7323 


.6936 


.72U3 


.7061 


.7081 


5 


60 


.6561 


.7547 


.6691 


.7431 


.6820 


.7313 


.6947 


.7193 


.7071 


.7071 





Cosine 


Sine 


v^obine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


Cosine 


Sine 


/ 


49° 1 


48° 1 


47° 1 


46° 1 45° 1 



TJ^BXjE II B, 






Natural Tangents and Cotangents. 




^ 


0° 


1° 


2° 


3° 


4" 1 


/ 
60 




Tang. jCotan. 


Tang. 


Cotan . 


Tang. 


Cotan . 


1 ang. 


Cotan . 


Tang. 


Cotan. 







.0000 Infinit. 


.0175 


57. 29 


.0349 


28.64 


.0524 


19.08 


.0699 


14.301 




5 


.0014 687.55 


.0189 


52.88 


.0363 


27.49 


0589 


18.56 


.0714 


14.008 


55 




10 


.0029 ,343 77 


.0204 


49.10 


.0378 


26.43 


.0533 


18.07 


.0728 


13.727 


50 




15 


.0044 


229.18 


.0218 


45.83 


.0393 


25.45 


.0568 


17.61 


.0743 


13.456 


45 




20 


0058 


171.88 


.0233 


42 66 


.0407 


21.54 


.0582 


17.17 


.0758 


13.197 40 




25 


.0073 


137.51 


.0247 


40.44 .0422 


23.69 


.0597 


16.75 


.0772 


12.947 35 




30 


0087 


114.59 


.0262 


38.19 


0437 


22.90 


.0612 


16.35 


.0787 


12.706 80 




35 


.0102 


98.22 


.0276 


36.18 


.0451 


22.16 


.0626 


15.97 


.0802 


12.474 


25 




40 


.0116 


85.94 


.0291 


34.37 


.0466 


21.47 


064J 


15.60 I 


.0816 


12.251 


20 




45 


.0131 


76.39 


.0305 


32.73 


.0480 


20.82 


.0655 


15.26 


.0831 


12.035 15 




50 


.0145 


68.75 


.0320 


31.24 


.0495 


20.21 


.0670 


14.92 


.0846 


11.826 


10 




55 


.0160 


62.50 


.0335 


29.88 


.0509 


19.63 


.0685 


14.61 


.0860 


11.625 


5 




60 


.0175 


27.29 


.0349 


28.64 


.0524 19.08 


.0699 


14.30 


.0875 


11.430 







Cotan 


Tane. 


Cotan, 


rang i 


V otan. 1 Tang. | 


Cotan. 


Tang. 


Cotan. 


Tang. 


f 




89° 


88° i 87° 1 


8t)° 


85° 




\ 


5° 


6° 


7° 


8° 


9° 






L'anid:. 


Cotan. 


1 ane. 


Cotan. 


Tang. 


Cotan. 


Tang. Cotan. 


Tang 


Cotan. 







.0875 


11.430 


.1051 


9.514 


.1228 


8.144 


.1405 7.115 


.1584 


6.314 


60 




5 


.0889 


11.242 


.1066 


9.383 


.1243 


8.047 


.1420 


7.041 


.1600 


6.255 


55 




10 


.0904 


1.053 


.1080 


9.255 


.1257 


7.953 


.1435 


6.968 


.1614 


6.197 


50 




15 


.0719 


10.883 


.1095 


9.131 


.1272 


7.860 


.1450 


6.897 


.1629 


6.140 


45 




20 


.0933 


10.712 


.1109 


9.010 


.1287 


7.770 


.1465 


6 827 


.1644 


6.084 


40 




25 


.0948 


10.546 


.1125 


8.892 


.1302 


7.682 


.1480 


6.758 


.1658 


6.030 


35 




30 


.0963 


10.385 


.1139 


8.777 


.1316 


7.596 


.1494 


6.691 


.1673 


5.976 


30 




35 


.0978 


10.229 


.1154 


8.665 


.1331 


7.511 


.1509 


6.625 


.1688 


5.923 


25 




40 


.0992 


10.078 


.1169 


8.555 


.1346 


7.429 


.1.524 


6.561 


.1703 


5.871 


20 




45 


.1007 


9.931 


.1184 


8.449 


.1361 


7.348 


.1539 


6.497 


.1718 


5.820 


15 




50 


.1022 


9.788 


.1198 


8.345 


.1376 


7.269 


.1554 


5.435 


.1733 


5.769 


10 




55 


.1036 


9.649 


.1213 


9.243 


.1391 


7.191 


.1569 


6.374 


.1748 


5.720 


5 




60 


.1051 


9.514 


.1228 


8.144 


1405 


7.115 


.1584 


6.314 


.1763 


5.671 







^ 


Cotan. 1 Tang;. 


Cotan. 


Tang. 


Cotan. 


lang. 


Cotan. 


ang 


Cotan. 


lang. 


f 




84° 


83° 


82° 


81° 


80° 




^ 


10° 


11° 


12° 


13° 


14° 


60 




Tang. |»^otan . 


Tang. |Cotan. 


Tang 


Cotan. 


Tang. 


Cota . 


1 ang 


Cotan . 







.1763 


5.671 


.1944 15 144 1 .2126 


4.705 


.2308 


4.331 


.2493 


4.011 




5 


.1778 


5.623 


.1959 


5.104 


.2141 


4.671 


.2324 


4.303 


.2509 


3.986 


55 




10 


.1799 


5.576 


.1974 


5.066 


.2156 


4.638 


.2339 


4.275 


.2524 


3.962 


50 




15 


.1808 


5.530 


.1989 


5.097 


.2171 


4.606 


.2355 


4.247 


.2540 


3.937 


45 




20 


.1823 


5.484 


.2004 


4.989 


.2186 


4.574 


.2370 


4.219 


.2555 


3.914 


40 




25 


.1838 


5.140 


.2019 


4.952 


.2202 


4.542 


.2385 


4.192 


.2571 


3.890 


35 




30 


.1853 5.395 


.2034 


4.915 


.2217 


4.511 


.2401 


4.165 


.2586 


3.867 


30 




35 


.1868 


5.352 


.2050 


4.879 


.2232 


4.480 


.2416 


4.139 


.2602 


3 844 


25 




40 


.1883 


5.309 


.2065 


4 843 


.2247 


4.449 


.2432 


4.113 


.2617 


3.821 


20 




45 


.1899 


5.267 


.2080 


4.808 


.2263 


4.419 


.2447 


4.087 


.2633 


3.798 


15 




50 


.1914 


5.226 


.2095 


4.773 


.2278 i 4.390 


.2462 


4.061 


2648 


3.776 


10 




55 


.1929 


5.185 


.2110 


4.738 


.2293 


4.360 


.2478 


4.036 


.2664 


3.754 


5 




60 


.1944 


5.144 


.2126 


4.705 


.2309 


4.331 


.2493 


4.011 


.2679 


3.732 







Cotan. 


Tang. 


Cotan. 


Tang 


Cotan. 


lang 


Cotan. 


Tang. 


Cotan. 


lang. 


.1 


79° 


78° 


77° 


76° 


75° ^ 




I 



T.A.BILS II B, 




Natural Tangents 


and Cotangents — Cont'd. 


^ 


15" 


lb° i 


17° 


18° 1 


19° 1 


' 


Tan?. jCotan. 


Tang. 


Cotan. 


Tang. 


Cotan. 


fang. 


Cotan . 


Tang. 


Cotan. 





.2679 


3.732 


.2867 


3.487 


.3057 


3.271 


.3249 


3.078 


.3443 


2.904 


30 


5 


.2695 


3 710 


.2883 


3.468 


.3073 


3 . 254 


3265 


3.062 


.3460 


2.890 


55 


1(1 


.2711 


3 689 


.2899 


3.449 


.3089 


3.237 


.3281 


3.017 


.3476 


2.877 


50 


15 


.2726 


3.6«8 


.2915 


3.431 


.3105 


3.220 


.3297 


3.033 


.3492 


2.864 


45 


•20 


.2742 


3.647 


.2930 


3 412 


.3121 


3.204 


.3314 


3.018 


.3.508 


2.850 


40 


25 


.2758 


3.626 


.2946 


3.394 


.3137 


3.188 


.3330 


3.003 


.3525 


2 . 837 


15 


30 


.2773 


3.606 


.2962 


3.376 


.3153 


3.172 


.3346 


2.989 


. 3551 


2.824 


^0 


35 


.2789 


3.5«6 


.2978 


3.358 


.3169 


3.1.56 


.3362 


2.974 


.35.58 


2.811 


25 


40 


.2805 


3.566 


.2994 3.340 


.3185 


3.140 


3378 


2.960 


.3574 


2.798 


20 


45 


.2820 


3.546 


.3010 3.323 


.3201 


3.124 


.3394 


2.946 


.3590 


2.785 


15 


50 


.2836 


3.526 


.3025 


3.305 


.3217 


3.108 


.3411 


2.932 


.3607 


2.772 


10 


55 


.2852 


3.507 


.3041 


3.288 


.3233 


3.093 


.3427 


2.918 


.3623 


2.760 


5 


60 


.2867 


3.487 


.3057 


3.271 


.3249 


3.078 


.3443 


2.904 


.3640 


2.747 





Cotan 


i'an . 


Cotan. 


Tang 


coian. 


Tang. 


Cotan. 


Tang. 


Cotan. 


I'ang. 


/ 


74° 1 


73° 


72° 1 


71° 


70° 


~0 


20° 1 


21° 


22° 


23° 


24° 


' 


Tana. 


Cotan. 


Pane. 


Cotan. 


Tang. Cotan. 


Tang. 


Cotan. 


Tang. 


Cotan. 


.3640 


2.747 


.3839 


2.605 


.4040 2.475 


.4245 


2.356 


.44.52 


2.246 


60 


5 


.3656 


2.735 


.3855 


2.594 


.4054 2.467 


.4258 


2.348 


.4470 


2.237 


55 


10 


.3673 


2.723 


.3872 


2.583 


.4074 2.454 


.4279 


2.337 


.4487 


2.229 


50 


15 


.3689 


2.711 


.3889 


2.571 


.4091 2.444 


.4296 


2.328 


.4505 


2.220 


45 


20 


.3706 


2.698 


.3905 


2.560 


.4108 2.434 


.4314 


2.318 


.4.522 


2.211 


40 


25 


.3722 


2.686 


.3922 


2.549 


.4125 2.424 


.4331 


2.309 


.4540 


2.201 


35 


30 


.3739 


2.675 


.3939 


2.539 


.4142 2.414 


.4348 


2.300 


.4557 


2.194 


30 


35 


.3755 


2.663 


.3956 


2.528 


.4159 


2.404 


.4365 


2.291 


.4575 


2.186 


25 


40 


.3772 


2.651 


.3973 


2.517 


.4176 


2.394 1 .4383 


2.282 


.4592 2.177 


20 


45 


.37e9 


2.639 


.3989 1 2.501 


.4193 


2.385 .4400 


2.273 


.4610 2.169 


15 


50 


.3805 


2.628 


.4006 


2.496 


.4210 


2.375 


.4417 


2.264 


.4628 2.161 


10 


55 


.3822 


2.616 


.4023 


2.485 


.4228 


2.365 


.4435 2.255 


.4645 2.153 


5 


60 


.3839 


2.695 


.4040 


2.475 


4245 


2.3.56 


.44.52 2.246 


.4663 2.144 





^ 


Cotan. i Tans. 


Cotan. 


Tang. 


Cotan. 


i ang, 


Coian. I ans 


Cotan. i'i'ang. 


69° 


68° 


67° 


66° 


65° 




~0 


25° 


26° 


27° 


28° 


29° 


/ 

60 


Tang. 


V otan. 


Tang. 


Cotan. 


Tang. 


Cotan. 


Tang. Cotan. 


i ang 


Cotan. 


.4663 


2.144 


.4877 


2 050 


.5095 


1 962 


.5317 


1.881 


..5.543 


1.804 


5 


.4681 


2.136 


.4895 


2.043 


.5114 


1.956 


..5336 


1.874 


.5562 


1.798 


55 


10 


.4698 


2.128 


.4913 


2.035 


..5132 


1.949 


.5354 


1.868 


.5.581 


1.792 


50 


15 


.4716 


2.120 


.4931 


2.028 


,5150 


1.942 


.5373 


1.861 


.5600 


1.786 


45 


20 


.4734 


2.112 


.4949 


2.020 


.5169 1.935 


.5392 


1.855 


.5619 


1.779 


40 


25 


.4752 


2.104 


.4968 


2.013 


.5187 


1.928 


.5411 


1.848 


.5638 


1.773 


35 


30 


.4770 


2.096 


.4986 


2.005 


.5206 


1.921 


.5429 


1.842 


.5658 


1.767 


30 


35 


.4788 


2.089 


.5004 1.998 


..5224 


1.914 


.5448 


1.835 


.5677 


1 761 


25 


40 


.4805 


2.081 


.5022 1 991 


.5243 


1 .907 ^ .5467 


1.829 


.5696 


1.755 


20 


45 


.4823 ' 2.073 


.5040 1.984 


.5261 


1.901 


.5486 


1.823 


.5715 


1.750 


15 


50 


.4841 2.065 


..50.59 1.977 


.5280 


1.894 


.5505 


1.816 


5735 


1.744 


10 


55 


.4849 2.058 .5077 1.970 


.5298 


1.887 


.5524 


1.810 


.5754 


1.738 


5 


60 


.4877 1 2.050 L -5095 | 1.963 


.5317 


1.881 


.5543 


1.804 


.5773 


1.732 





Jotan. 1 Tang. 


Cotan. 1 Tang 


Cotan. I 1 ang 


Cotan. 


Tang, 


Cotan 


iang. 


64° 


63° 


62° 


61° 


60° 



T^BILIE II B, 


Natural Tangents and Cotangents — Cont'd. 


> 

"o 


30° 


3i° 


3^° 


33" 


ii'^ 


/ 


Tang. Cotan. 


Tang. 


Cotan. 


Tang. 


Cotan. 


Tang. 


Cotan. 


Tans, 


Cotan. 


.5773 


1.732 


.6008 


1.664 


.62*9 


1.600 


.6494 


1.540 


.6745 


1.483 


60 


5 


.5793 


1.726 


.6028 


1.659 


.6269 


1.595 


6515 


1.535 


.6766 


1.478 


55 


10 


.5812 


1,720 


.6048 


1.653 


.6289 


1.590 


.6535 


1.530 


.6787 


1.473 


50 


15 


.5832 1.715 


.6068 


1.648 


.6309 


1.5S5 


.6556 


1.525 


.6809 


1.469 


45 


20 


5851 '1.709 


.6088 


1 643 


.6330 


1.580 


.6577 


1.520 


.6830 


1.464 


40 


25 


.5871 


1.703 


.6108 


1.637 


.63-30 


1.575 


.6598 


1.516 


.68.51 1.4601 


35 


30 


5891 


1.698 


.6128 


1.632 


.6-^71 


1.570 


.6619 


1.511 


.6873 


1.455 


30 


35 


.5910 


1.692 


.6148 


1.626 


.6391 


1.565 


.6640 


1.506 


.6894 


1.4.50 


25 


40 


.5930 


1.686 


.6168 


1.621 


.6-12 


1.560 


6661 


1.501 


.6916 


1.446 


20 


45 


.5949 


1.681 


.6188 


1.616 


.64^*2 


1.555 


.6682 


1.497 


.6937 


1.441 


15 


50 


.5969 


1.675 


.6208 


1.611 


.6-^53 


1.550 


.6703 


1.492 


.6959 


1.438 


10 


55 


.5989 


1.670 


.6228 


1.606 


.6^73 


1.545 .6724 


1.487 


.6980 


1.433 


5 


60 


.6099 


1.664 


.6249 


1.600 


.6494 


1..540 .6745 


1.483 


.7002 


1.428 


_0 


Cotan 


lan^. 


Cotan. 


Tang 


-,oian. 


1 ang. Cotan. 


Tang. 


i^otan. 


Tang. 




59° 


58° 


57° 1 56° 


55° 


^ 


- 35° 


36° 


37° 


38° 


39° 


/ 


Tana. 


Cotan. 


Tane. 


Cotan. 


Tang. 


Cotan. 


Tang. 


Cotan. 


Tang. 


Cotan. 





.7002 


1.428 


.7265 


1.376 


.7535 


1.327 


.7813 


1 288 


.8098 


1.235 


60 


5 


.7024 


1.424 


.7288 


1.372 


.7558 


1.323 


.7836 


1.276 


.8122 


1.231 


55 


10 


.7045 


1.419 


.7310 


1.368 


.7581 


1.319 


.7860 


1.272 


.8146 


1.228 


50 


15 


.7067 


1.415 


.7332 


1.364 


.7604 


1.315 


.7883 


1.268 


.8170 


1.224 


45 


20 


.7089 


1.411 


.7355 


1.360 


.7627 


1.311 


.7907 


1.265 


.8195 


1.220 


40 


25 


.7111 


1.406 


.7377 


1.355 


.7650 


1.307 


.7931 


1.261 


.8219 


1.217 


35 


3C 


.7133 


1.402 


.7400 


1.351 


.7673 


1.303 


.7954 


1.257 


.8243 


1.213 


30 


35 


.7155 


1.398 


.7422 


1.347 


.7696 


1.299 


.7978 


1.2.53 


.8268 


1.209 


25 


40 


.7178 


1.393 


.7445 


1.343 


.7719 


1.295 


.8002 


1.250 


.8292 


1.206 


20 


45 


.7199 


1.389 


.7467 


1.339 


.7742 


1.291 


.e026 


1.246 


.8317 


1.202 


15 


50 


.7221 


1.385 


.7490 


1.335 


. 7766 


1.287 


.8050 


1.242 


.8341 


1.199 


10 


55 


.7243 


1.381 


.7513 


1.331 


.7789 1.284 


.8074 


1.239 


.8366 


1.195 


5 


60 


.7265 


1.376 


.7535 


1.327 


7813 1.280 


.8098 


1.235 


.8391 


1.192 


J) 


> 


Cotan. 


Tang. 


Cotan. 


Tang. 


Cotan. 1 ang, 


Cotan. 


1 ang 


Cotan. 


lang. 


54° 


53° 1 


52° 


5 


o 


50° 


^ 


40° 


41^ 


42° 


43° 


44° 


/ 
60 


Tang. 


otan. 


Tang. 


Cotan. 


Tang. 


Cotan. 


Tang, 


Cota . 


1 ang 


Cotan. 





.8391 


1.192 


.8693 


1 150 


.9004 


1 111 


.9325 


1.072 ^ .9657 


1.036 


5 


.8416 


1.188 


.8718 


1.147 


.9030 


1.107 


.9352 


1.069 


.9685 


1.032 


55 


10 


.8441 


1.185 


.8744 


1.144 


.90.57 


1.104 


.9380 


1.066 


.9713 


1.029 


50 


15 


.8466 


1.181 


.8770 


1.140 


,9083 


1.101 


.9407 


1.063 


.9742 


1.026 


45 


20 


.8491 


1.178 


.8795 


1.137 


.9101 


1.098 


.9434 


1.060 


.9770 


1.023 


40 


25 


.8516 


1.174 


.8821 


1.134 


.9137 


1.094 


.6462 


1.057 


.9798 


1.021 


35 


'Sd 


.8541 


1.171 


.8847 


1.130 


.9163 


1.091 


.9490 


1.0.54 


.9827 


1.018 


30 


35 


.8566 


l.]67 


.8873 


1.127 


.9190 


1.088 


.9517 


1.051 


.9856 


1 015 


25 


40 


.8591 


1.164 


.8899 


1 124 


.9217 


1.085 


.9545 


1.048 


.9884 


1.012 


20 


45 


.8617 


1.161 


.8925 


1.120 


.9244 


1.082 


.9573 


1.045 


.9913 


1.009 


15 


50 


.8642 


1.157 


.89.51 


1.117 


.9271 


1.079 


.9601 


1.042 


.9942 


1.006 


10 


55 


.8667 


1.154 


.8978 


1.114 


.9298 


1.075 


.9629 


1.039 


.9971 


1.003 


5 


60 


.8693 


1.150 


.9004 


1.111 


.9325 


1.072 


.9657 


1.036 


1.000 


1.000 


_0 
/ 


Cotan. 


Tang. 


Cotan. 


Tang 


Cotan. 


lang 


Cotan. 


rang. 


Cotan 


Tang. 


49° 


48° 


47° 


46° 


45° 



T-i^BIjIE II C, 


Natural Secants and Cosecants 


1) 


0- 


1° 


i" 


3° 


4° 


^ 


Secant 


v^osec. 


Se^ani 


Cosec. 


Secant| Cosec. 


Secant! Cosec. 


Secantj Coj^ec. 


1.0000 


Infinit 


1 0001 


57.300 


1.0006128.654 


1.0014J19.107 


1. 002414. 335 


60 


5 


1.0000 


687.55 


1.00i)2 


52.891 


1.0007127.508 


1.001418.591 


1.002514.043 


55 


10 


1.0000 


343.77 


1.0002 


49.114 


1.0(K)7]26.150 


1.001518.103 


1.O026 13.763 


50 


15 


1.0000 


229.18 


1.0002 


45.840 


1.000825.471 


1.001617.639 


1.0027il3.494 


45 


•20 


1 0000 


171. «9 


1.0003 


42.976 


1.000S24..562 


1.0017:17.198 


1.0029,13.255 


40 


25 


1 0080 


137.51 


1.0003 


40.448 


1.0099 23.716 


1.0018116.779 


1.003012.985 


35 


30 


1.0000 


114 59 


1.0003 


38.201 


1.000922.925 


1.0019:16.380 


1.003112.745 


30 


35 


1.0000 


98.22 


1.0004 


36.191 


1.6010 22.186 


1.00J9;i6.000 


1.003212.514 


25 


40 


1.0001 


85.94 


1.0004 


34.382 


1 001121.494 


1.0020;15.637 


1.003312.291 


20 


45 


1.0001 


76.39 


1.9005 


32. 745 


1.001120 843 


1.0021115 290 


1.003412.076 


15 


50 


1.0001 


68.76 


1.0005 


31.257 


1.0012 20.230 


1.0022!l4.958 


1.003611.868 


10 


55 


1.0001 


62.51 


1.0005 


29 899 


1.0013 


19.653 


1.0023 


14.«40 


1.0037111.668 


5 


60 


1.0601 


57.30 


1.0006 


28.654 


1.0014 


19.107 


1.0024 


14.335 


1 003811.474 





Cosec. 


Secant 


Cosec 


Secant 


Cosec. 


Secant 


Cosec. 


-lecani 


Cosec. S cant 


' 


8S 


o 


88° 1 


87° 


86° 1 


85° 


^ 


5° 


6° 


7° 


b° 1 


9° 


60 


Secant 


Cosec. 


Secant 


Cosec, 


Secant 


Cosec 


Secant 


Cosine 


Secant 


Cosec 





1.0038 


11.474 


1.0055 


9.567 


1.0075 


8.205 


1.0098 


7.185 


1 0125 


6.392 


5 


1.0039 


11.286 


1.0057 


9.436 


1.0077 


8.109 


1.0100 


7.112 


1.0127 


6 334 


55 
50 


10 


1.0041 


11.104 


1.0058 


9.309 


1.0079 


8.016 


1 0102 


7 039 


1.0129 


6.277 


15 


1.0042 


10.929 


1.0060 


9.185 


1.0081' 


7.924 


1.0104 


6.969 


1.0132 


6.221 


45 


20 


1.0043 


10.758 


1.0061 


9. 065 


1.0082 


7.«34 


1.0107 


6 900 


1. 0134 


6.166 


40 


25 


1.0045 


10 593 


1.0063 


8.948 


1.0084 


7.747 


1.0109 


6 832 


1.0136 


6.112 


35 


30 


1.0046 


10.433 


1.0065 


8.834 


1.0086 


7.661 


1.0111 


6.765 


1.0l.i9 


6.059 


30 


35 


1.0048 


10.278 


1.0066 


8.722 


1.0088 


7.577 


1.0113 


6.700 


1.0141 


6.007 


25 


4C 


1.0049 


10.127 


1.0068 


8.614 


1.0090 


7.496 


1.0115 


6.636 


1.0144 


5.955 


20 


45 


1.0050 


9.981 


1.0070 


8.508 


1.0092 


7.416 


1.0118 


6.574 


1.0146 


5.905 


15 


50\ 


1.0052 


9.839 


1.0071 


8.405 


1.0094 


7.337 


1.0120 


6.512 


1 0149 


5.855 


10 


55 


1.0053 


9.701 


1.0073 


8.304 


1.0096 


7.260 


1.0122 


6.452 


1.0152 


5.807 


5 


60 


1.0055 


9.567 


1.0075 


8.205 


1.0098 


7.185 


1.0125 


6 392 


1.0154 


5.7.59 





^ 


Cosec. 


Secant 


Cosec. 


Secant 


Cosec. 


Secant 


Cosec. 


Secant 


Cosec. iSecant 


' 


84° 


83° 


82° 


81° 


80° 


^ 


10° 


11° 


12° 1 13° 


14° 


60 


Secant 


Cosec. 


Secant 


Co<ec 


Secant 


Cosec 


Secant 


Cosec. 


Secant 


V ijsec. 





1.0154 


5.789 


1.0187 


5.241 


1.0223 


4.810 


1.0263 


4.445 


1.0306 


4.134 


5 


1.0157 


5.712 


1.0190 


5. 202 


1.0226 


4.777 


1.0266 


4.417 


1.0310 


4.109 


55 


10 


1.0159 


5.665 


1.0193 


5.164 


1.0230 


4.745 


1.0270 


4 390 


1.0314 


4.086 


50 


15 


1.0162 


5.620 


i.ort6 


5.126 


1.0233 


4.713 


1.0273 


4.363 


1.0317 


4. 062 


45 


20 


1.0165 


5.574 


1.0199 


5.088 


1.0236 


4.682 


1. 0277 


4.336 


1.0321 


4.039 


40 


25 


1.0167 


5.531 


1 0202 


5.052 


1.0239 


4. 651 


1.0280 


4.310 


1.0325 


4.016 


35 


30 


1.0170 


5.487 


1.0205 


5.016 


1.0243 


4.620 


1.0284 


4.284 


1.0329 


3.994 


30 


35 


1.0173 


5.445 


1.0208 


4.980 


1.0246 


4.590 


1.0288 


4.258 


1.0333 


3.9^2 


25 


40 


1. 0176 


5.403 


1.0211 


4.945 


1.0249 


4.560 


1.0291 


4.232 


1.0337 


3.949 


20 


45 


1.0179 


5.361 


1.0214 


4.911 


1.02.53 


4 . 531 


1.0295 


4.207 


1.0341 


3.828 


15 


50 


1.0181 


5 320 


1.0217 


4.876 


1.0256 


4.502 


1.0300 


4.182 


1.0345 


3.906 


10 


55 


1.0184 


2.280 


1.0220 


4.843 


1.0260 


4.474 


1. 0302 


4.158 


1.0349 


3.845 


5 


60 


1.0187 


5.241 


1.0223 


4.810 


1.0263 


4.445 


1 0306 


4.134 


1.0353 


3.864 





Cosec 


S'^cant 


Cosec. 


Secant 


Cosec 


Secant 


Cose 


'^ecant 


Cos«c. 


Seran' 


79° 


78° 


77° 


76° 1 75° 



TJ^SXjE II c. 


Natural Secants and Cosecants — Continued. 


> 


16" 


lb° 


17° 


18° 


19^ 


' 


Secant 


Cosec. 


Secant! Cosec. 


Secant 


Co^ec. 


recant 


Cosec. 


^secant 


Cosec. 





1.0353 


3.864 


1.0403 3.628 


1.0457 


3.420 


1.0515 


3.236 


1.0576 


3.071 


60 


5 


1.0357 


3.843 


1.04'^"'; 3.610 


1.0461 


3 . 404 


1.0520 


3.222 


1.U.581 


3.059 


55 


10 


1.0361 


3.822 


1.0412 


3.591 


1.0466 


3.388 


1.0525 


3.207 


1.0587 


3.046 


50 


15 


1.0365 


3.802 


1.0416 


3.574 


1..0471 


3.372 


1.0530 


3.193 


1.0592 


3.033 45 1 


20 


].0369 


3.782 


1.0420 


3.. 5.56 


1.0476 


3.356 


1.0535 


3.179 


l..C'J8 


3.021 


40 


25 


1.0374 


3.762 


1.0425 


3.538 


1.0480 


3.341 


1 . 0540 


3.165 


1 . o603 


3.008 


35 


30 


1.0377 


3.742 


1.0429 


3.. 521 


1.04^5 


3.325 


1.0545 


3.151 


1.0608 


2.996 


30 


35 


1.0382 


3.722 


1.0134 


3.. 504 


1.0490 


3.310 


1.0550 


3.138 


1.0614 


2.983 


25 


40 


1.0386 


3.703 


1.0438 


3.487 


1.0495 


3.295 


1.0555 


3.124 


1.0619 


2.971 


20 


45 


1.0390 


3.684 


1.0443 


3.470 


1.0500 


3.280 


1.0560 


3.111 


1.0625 


2.959 


15 


50 


1.0394 


3.665 


1.0448 


3.453 


1.0505 


3.265 


1.0566 


3.098 


1.0630 2.947 


10 


55 


1.0399 


3.646 


1.0452 


3.437 


1.0510 


3.251 


1.0571 


3.085 


1.0636' 2.935 


5 


60 


1.0403 


3.628 


1.0457 


3.420 


1.0515 


3.236 


1.0576 3.071 


1.0642 2.924 





\ 


Cosec. 


Secant 


Cosec. 


Secant 


Cosec. 


Secant 


Cosec. jSecant 


Cosec. Ibecant 


' 


7^ 


1° 


73° 


72° 


71° 


70° 


1) 


20° 


21° 


22° 


23° 1 24° 


6(> 


Secant 


Cosec. 


Secant 


Cosec. 


Secant 


Cosec. 


Secant 


Cosec 


becant 


Cosec. 


1.0642 


2.924 


1.0711 


2.790 


1.0785 


2.669 


1.0864 


2.559 


1.0946 


2.459 


5 


1.0647 


2.912 


1.0717 


2.780 '1.0792 


2.660 


1.0870 


2.551 


1.0953 


2.451 


55 


10 


1.0653 


2.901 


1.0723 


2.769 1.0798 


2.650 


1.0877 


2.542 


1.0961 


2.443 


50 


15 


1.0659 


2.889 


1.0729 


2.759 


1.0804 


2.641 


1.0^84 


2.533 


1.0968 


2.435 


45 


20 


1.0664 


2.878 


1.0736 


2.749 


1.0811 


2.632 


1.0891 


2.525 


1.0975 


2.427 


40 


25 


1.0670 


2.867 


1.0742 


2.738 


1.0817 


2.622 


1.0897 


2.516 


1.0982 


2.419 


35 


30 


1.0676 


2.855 


1.0748 


2.728 


1.0824 


2.613 


1.0904 


2.508 


1.0989 


2..411 


30 


35 


1.0682 


2.844 


1.0754 


2.718 


1.0830 


2.604 


1.0911 


2.499 


1.0997 


2.404 


25 


40 


1.0688 


2.833 


1.0761 


2.708 


1.0837 


2.595 


1.0918 


2.491 


1.1004 


2.396 


20 


45 


1.0694 


2.822 


1.0766 


2.698 


1.0844 


2.586 


1.0925 


2.483 


1.1011 


2.389 


15 


50 


1.069912.812 


1.0773 


2.689 


1.0850 


2.577 


1.0932 


2.475 


1.1019 


2.381 


10 


55 


1.0705 2.801 


1.0779 


2.679 


1.0857 


2.568 


1.0939 


2.467 


1.1026 2.374 


5 


60 


1.0711[ 2.790 


1.07851 2.669 


1.0864 2.5.59 


1.0946 


2 459 


1.1034,2.366 





Cosec. jSecant 


Cosec. Secant 


Cosec. iSecant 


Cosec, 


Secant 


Cosec. ISecant 


69^ 


68° 


67° 1 66° 


65° 


^ 


25° 


26° 


27° 


28° 


2t.° 


/ 


Secant 


Cosec 


Secant 


Cosec 


Secantj Cosec. 


Secant 


Cosec. 


Secant 


Cosec. 





1.1034 


2.366 


1.1126 


2.281 


1.1223 


2.203 


1.1326 


2.130 


1.1433 


2.063 


60 


5 1.1041 


2.359 


1.1134 


2.274 


1.1231 


2.196 


1.1334 


2.124 


1.1443 


2.057 


55 


10 1.1019 


2 351 


1.1142 


2.267 


1.1240 


2.190 


1.1343 


2.118 


1.1452 


2.052 


50 


15 1.1056 


2.344 


1.1150 


2.261 


1.1248 


2.184 


1.1352 


2.113 


1.1461 


2.047 


45 


2011.1064 


2.337 


1.1158 


2.254 


1.1257 


2.178 


1.1361 


2.107 


1.1471 


2.041 


40 


25 1.1072 


2.330 


1.1166 


2.248 


1.1265 


2.172 ,1.1370 


2.101 : 1.1480 


2.036 


.35 


30| 1.1079 


2.323 


1.1174 


2.241 


1.1274 


2.166 


1.1379 


2.(96 i 1.1489 


2.031 


30 


35 1.1087 


2.316 


1.1182 


2.235 


1.1282 


2.160 


1.1388 


2.090 ; 1.1499 


2.026 


25 


40 1.1095 


2.309 


L.1190 


2.228 


1.1291 


2.154 


1.1397 


2.085 i 1.1508 


2.020 


90 


45 1.1102 
5rt! 1.1110 
55 1.1118 


2.302 


1.1198 


2.222 


1.1299 


2.148 


1.1406 


2.679 1.1518 


2.015 


15 


2.295 


1.1207 


2.215 


1.1308 


2.142 


1.1415 


2.073 1.1528 


2.010 


10 


2.288 


1.1215 


2.209 


1.1317 


2.136 


1.1424 


2.068 1 1.15371 2.005 


5 


60 


1.1126 


2.281 


1.1223 


2 203 


1.1326 


2.130 


1.1433 


2.063 1.1.547i 2.000 





^ 


Cosec. 


recant 


Cosec. 


Secant 


Cosec. 


Secant 


Cosec. 


Secant 


Cosec.lSecant 


L 


64° 1 63° 1 


62° 


6 ° 


60° 1 



TJ^BIjE ii c. 


Natural Secants and Cosecants — Continued. 


1) 


30" 


31° 


32^ 


133° 


34° 


60 


Secant 


Cosec 


SecantlCosec 


Secant Cosec. 


Secant 


Cosec. 


Secantl Cosec. 


1.1547 


2.000 


1 1666 1.942 


1.1792 1.887 


1.J924 


1.836 


1.2062 1.788 


5 


1 1557 


1.995 


1.167611.937 


1.1802 1.883 


1.1935 


1.832 


1.2074^ 1.784 


55 


10 


1.1566 


1.990 


1.1687 


1.932 1.1813 


1.878 


1.1946 


1.8-28 


1.2086: 1.781 


50 


15 


].1576 


1.985 


1.1697 


1.928 1.1824 


1.874 


1.1958 


1.824 


1.2098 1.777 


45 


'20 


1 1586 


1.9^0 


1.1707 


1.923 


1 1835 


1.870 


1.1969 


1.820 


1.2110 1.773 


40 


25 


1 1596 


1.975 


1.1718 


1.918 


1.1846 


1.865 


1.1980 


1.816 


1.2122 1.769 


35 


80 


1.1606 


1 970 


1.1728 


1.914 


1.1857 


1.861 


1.1992 


1.812 


1.2134; 1.765 


30 


35 


1.1616 


1.965 


1.1739 


1. 909 


1.1868 


1.857 


1.2004 


1.808 


1.2146; 1.762 


25 


40 


1.1626 


1.9 1 


1.1749 


1.905 


1 1879 


1 853 


1.2015 


1.804 


1.21.58 


1.7.58 


20 


45 


1.1636 


1.956 


1.1760 


1.900 


1.1890 


1 848 


1.2^)27 


1 800 


1.2171 


1.754 


1§ 


50 


1 . 1646 


1.951 


1.1770 


1 896 


1.1901 


1.844 


1.2039 


1.796 


1.2183 


1.751 


10 


55 


1 1656 


1.946 


1.1781 


1.891 


1.1912 


1.940 


1.2050 


1.792 


1.2195 


1.747 


5 


6U 


1.1666 


1.942 


1.1792 


1.887 


1.1924 


1.836 


1.2062 


1.788 


1 2208 


1.743 


^ 


Cosec. 


Secant 


Cosec 


Secant 


< osec. 


Secant 


Cosec. 


■^ec.ini 


Cosec. 


S- cant 


5S 


58° 1 


57° 1 56° 


55° 


"o 


3o° 


36° 


37° 


38° 


39° 


60 


S -cant 


Cosec. 


Secant 


Cos-c, 


Secant 


Cosec 


>ecant 


Cospc 


Secant 


Cosec 


1.2208 


1.743 


1.2361 


1.701 


1.2521 


1.662 


1.2690 


1.624 


1.2867 


1.589 


5 


1.1^2 


1.740 


1 2374 


1.698 


1.2535 


1 6.58 


1.2705 


1.621 


1.2883 


1 586 


55 


10 


1.2233 


1.736 


1.2387 


1 694 


12549 


1 655 


1 2719 


1 618 


1.2898 


1.583 


50 


15 


1.2245 


1.733 


1.2400 


1.691 


1.2563 


1 652 


1.2734 


1.615 


1.2913 


1.580 


45 


20 


1.2258 


1 . 72iJ 


1.2413 


1.688 


1.2577 


1 649 


1.2748 


1 612 


1. 2929 


1.578 


40 


25 


1.2270 


1 726 


1.2427 


1.684 


1.2501 


1.646 


1.2763 


1 609 


1.2944 


1 575 


35 


30 


1.2283 


1.722 


1.2440 


1.6^1 


12005 


1.643 


1.277« 


1.6r.6 


1.2960 


1.572 


30 


35 


1.2296 


1.718 


1.2453 


1.678 


1.2619 


1.640 


1.2793 


1.603 


1.2975 


1.569 


25 


40 


1.2309 


1715 


1.2467 


1 675 


1 . 2633 


1.636 


1.2807 


1.600 


1.2991 


1.567 


20 


45 


1.2322 


1.7J2 


1.2480 


1 671 


1.2f^47 


1.633 


1.2822 


1.598 


1.3006i 1..564 


15 


50 


1.2335 


1.708 


1 2494 


1 668 


1.2661 


1.630 


1.2837 


1.595 


1 3022 1.561 


10 


55 


1.2348 


1.705 


1.2508 


1.665 


1.2676 


1.627 


1.2852 


1.592 


1.3038; 1 558 


5 


60 


1 2364 


1.701 


1.2521 


1.662 


1.2690 


1.624 


1.2867 


1.589 


1.30.54 


1..556 

Secant 


J) 


Cosec. 


Secant 


Coser. 


Secant 


Cosec. 


Secant 


Cosec. 


Secant 


Cosec. 


5-1° 1 


53° 1 


52° 


51° 


50° 


1, 


40° 


41° 


42° 


43° 


44° 


/ 

60 


Secant 


Cosec 


Secant 


Co ec 


Secant 


Cosec 


Secant 


Cosec 


Secant 


V osec. 


1.3054 


1.556 


1.325 


1.524 


1.346 


1.4^>4 


1.367 


1.466 


1.390 


1.439 


5 


1.3070 


1.553 


1.327 


1 522 


1.347 


1 492 


1.369 


1.464 


1.392 


1.437 


55 


10 


1.3086 


1.550 


1.328 


1.519 


1.349 


1.490 


1.371 


1.462 1.394 


1.435 


50 


15 


1.3102 


1.548 


1 .330 


1.517 


1.351 


1.487 


1.373 


1.459! 1.396 


1.433 


45 


20 


1.3118 


1.545 


1.332 


1.514 


1.353 


1.485 


1 375 


1.457 


1.398 


1.431 


^0 


25 


1.3134 


l.o42 


1 333 


1.512 


1.354 


1.483 


1.377 


1.455 


1.400 


1.429 


35 


30 


1.3151 


1.539 


1.335 


1.509 


1.356 


1.480 


1.379 


1.453 


1 .402 


1.427 


30 


3;- 


1.3167 


1.537 


1.337 


1.507 1.358 


1.478 


1.381 


1. 450 


1.404 


1.425 


25 


40 


1.3184 


1.534 


1.339 


1.504 ! 1.360 


1.476 


1.382 


1.448 


1.406 


1.423 


20 


45 


1.3200 


'1 532 


1.340 


1.502 1.362 


1.473 


1.384 


1.446 


1.408 


1.420 


15 


50 


1.3217 


1 529 


1.342 


1 499 1.364 


1.471 


1.386 


1.444 


1.410 


1.418 


10 


55 


1.3233 


1.527 


1.344 


1.497 1.365 


1.469 


1.388 


1.442 


1.412 , 1.416 


5 


60 


1.3250 


1.524 


1.346 


1.494 1.367 


1 466 


1 390 


1.439 


1.414 1.414 




Cosec 


Secant 


Cosec. 


-ecant 


Cosec. 


Secant 


Cose . 


Secant 


Cos-c. Secant 


49° 


48° 


47° 


46° 


45° 



Trigonometrical and Conversion Table. III. 



BEING A TABLE OF RHUMBS. 



Name 

of 
Course. 



N. 
and 
S. 



Points. 






Degrees 
and 

Minutes, 



1 

49 
13 

373^ 

2 
26 
50M 



Sine 

or 

D eparture 



.0000 
.0246 
.0491 
.0735 
.0980 
.1224 
.1467 
.1709 



Co Sine 

or 
Diff. Lat. 



1.0000 
.9997 
.9988 
.9973 
.99;!;2 
.9925 
.9892 
.9853 



Tangent. 



.0000 
.0245 
.0492 
.0737 
.0985 
.1234 
.1483 
.1736 



Secant. 



0.0000 
1.0003 
1.0012 
1.0027 
1.0048 
1.0076 
1.0109 
1.0150 



and 
3.by|E,. 



N. N,{E-. 



and 



S. S. 



(E. 

iw. 



fE. byN. 
•tw.byN. 

and 

c fE. by S. 
^- 1 W. by S. 



% 

78 






5/8 

'A 



11 15 

12 395^ 

14 4 

15 28 

16 52M 

18 17 

19 41 
21 oK 



.1951 
.2191 
.2430 
.2667 
.2903 
.3137 
.3368 
.3599 



.9757 
.9700 
.9638 
.9569 
.9495 
.9417 
.9330 



22 30 

23 54^ 

25 19 

26 43 

28 VA 

29 32 

30 56 
32 20^ 



.3827 
.4053 
.4276 
.4496 
.4714 
.4929 
.5140 
.5350 



.9239 
.9142 
.9040 
.8932 
8819 
.8701 
.8578 
.8448 



33 


45 


.5556 


35 


9/2 


.5758 


36 


34 


.5958 


37 


58 


.6152 


39 


12% 


.6344 


40 


47 


.6532 


42 


11 


.6715 


43 


35K 


.6895 



.8315 
.8176 
.8032 
.7884 
.7730 
.7572 
.7410 
.7243 



.1989 
.2246 
.2506 
.2767 
.3034 
.3304 
.3577 
.3857 



.4142 
.4433 
.4730 
.5033 
.5345 
.5665 
. 5992 
.6332 



.6682 
.7043 
.7418 
.7803 
.8207 
.8627 
.9062 
.9520 



1.0196 
1.0249 
1.0309 
1 0376 
1.0450 
1.0532 
1.0629 
1.0718 



1.0824 
1.0939 
1.1062 
1.1195 
1.1339 
1.1493 
1.1658 
1.1836 



1.2025 
1.2231 
1.2451 
1.2684 
1 .2936 
1.3207 
1.3495 
1.380T 



122 



Trigonometrical and Conversion Table, ill. 

BEING A TABLE OF RHUMBS. 


Name 

of 
Course. 


Points. 


Degrees Sine 

and or 
INIinutes. Departure 


Co Sine 

or 
DifF. Lat. 


Tangent. 


Secant. 


X' f E. 
and 


4. 

3/8 


45 

46 24^ 

47 49 

49 13 

50 37K 

52 2 

53 26 

54 50% 


.7071 
.7242 
.7410 
.7572 
.7730 
.7884 
.8032 
.8176 


.7071 

.6895 
.6715 
.6532 
.6344 
.6152 
.5958 
.5758 


1.0000 
1.0,504 
1.1035 
1.1592 
1.2185 
1.2815 
1.3481 
1.4198 


1,4142 

1.4502 
1.4892 
1.5.309 
1..5763 
1.6255 
1.6785 
1.7366 


^, fE.byE. 
N'lw.byW. 

and 

^ fE.byE. 
^•\W, byW. 


5. 

/8 


56 15 

57 39% 

59 4 

60 28 

61 523^ 

63 17 

64 41 
66 bV^ 


.8315 
.8449 
.8578 
.8701 
.8819 
.8932 
.9040 
.9142 


.5556 
.5350 
.5i40 
.4929 
.4714 
.4497 
.4476 
.4053 


1.4966 
1.5793 
1.6687 
1 . 7651 
1.8708 
1.9868 
2.1139 
2.2558 


1.7999 
1.8694 
1.9454 
2.0287 
2.1214 
2. 2243 
2.3385 
2.4675 


E. N. E. 
W. N. W. 

and 

E. S.E. 
W. S. W. 


6. 


67 30 

68 54^ 

70 19 

71 43 

73 7^ 

74 32 

75 56 
77 20!x^ 


.9239 
.9330 
.9415 

.9495 
.9569 
.9638 
.9700 
.9757 


.3827 
.3599 
.3368 
.3137 
.2903 
.2667 
.2430 
.2191 


2.4142 
2.5926 
2.79.54 
3.0267 
3.2966 
3.6140 
3.9910 
4.4525 


2.6131 
2.7788 
2.9689 
3.18'^6 
3. 4449 
3.7498 
4.1144 
4.5634 


w.byN. 

and 

w! by s. 


7. 

3/8 

/8 
3/ 
% 


78 45 

80 91 

81 34 

82 58 

84 223^ 

85 47 

87 11 

88 353^ 


.9808 
.9853 
.9892 
.9925 
.9952 
.9973 
.9988 
.9997 


.1951 
.1709 
.1467 
.1224 

.0980 
.0735 
.0491 
.0246 


5.0273 5.1258 
5.7647 i 5.8505 
6.7448 6.8i86 
8.1054 ' 8.1668 
10.154 10.2023 
13.563 13.6002 
20.. 325 20.3809 
40.688 40.6889 


East— West. 8. 


90 00 1.0000 


.0000 


Infinite. 



123 



The Sun's Amplitude. IV. 

With rate of change of Azimuth, from Sunrise to the time of his crossing the 
Prime Vertical, and with the time of his Risinij and Setting. 



Lat. 


41° 


42^ 


43° 


44° 


45° 


Dec. 


ChHiige 

of Azimuth 

1^ in b^4 m. 


Change 

of Azimuth 

1° in 6}i m. 


Chancre 
of Azimuth 
1° in 6 m. 


Change 
of Azimuth 
1° in 6 m. 


Change 
of Azimuth 
1^ in 6 m. 




AMP. 


H. M. 


AMP. 


H. M. 


AMP. 


H. M. 


AMP. 


H. M. 


AMP. 


H. M. 


1° 


1°. 20' 


R5.57 
b 6.03 


i°.2r 


R5.56 

S6.04 


1°.22' 


R5.56 
=>6.04 


1^.23' 


R5.56 
S6.04 


1°.25' 


R5.56 
S6.04 


2 


2.39 


5 53 
6.07 


2.42 


5.53 
6.07 


2,44 


5.53 
6.07 


2.47 


5.52 
6.08 


2.50 


5.52 
6.08 


3 


3.59 


5.50 
6.10 


4.02 


5.49 
6.11 


4.06 


5.49 
6.11 


4.10 


5.48 
6.12 


4.15 


5 48 
6.12 


4 


5.18 


5.46 
6.14 


5.23 


5.46 
6.14 


5.28 


5-45 
6.15 


5.34 


5.45 
6.15 


5.40 


5.44 
6.16 


5 


6.38 


5 43 
6.17 


6.44 


5.42 
6.J8 


6.51 


5.41 
6.19 


6.58 


5.41 
6.19 


7.05 


5.40 
6.20 


6 


7.58 


5.39 
6.21 


8.05 


5.38 
6.22 


8.13 


5.38 
6.22 


8.21 


5.37 
6.23 


8.30 


5 36 

6.24 


7 


9.18 


5.35 
6.25 


9.26 


5.35 
6.25 


9.36 


5.34 
6.26 


9.45 


5.33 
6.27 


9.55 


5.32 

6.28 


8 


10.38 


5.32 

6.28 


10.48 


0.31 

6.29 


10.58 


5.30 
6.30 


11.09 


5.29 
6 31 


11.21 


5.28 
6.32 


9 


11.58 


5.28 
6.32 


12.09 


5.27 
6.33 


12 21 


5.26 
6.34 


12.34 


5.25 
6.35 


12.47 


5.24 
6.36 


10 


13.18 


5.25 
6.35 


13.31 


5.23 

6 37 


13.44 


5.22 
6.38 


13.58 


0.21 
6.39 


14.13 


5.19 
6.41 


11 


U.39 


5.21 

6.39 


14.53 


5.20 
6.40 


15.07 


5.18 
6.42 


15.23 


5.17 
6.43 


15.39 


5.15 
6.45 


12 


15.59 


5.17 
6.43 


16.15 


5 16 
6.44 


16.31 


5.14 
6.46 


16.48 


5.13 

6.47 


17.06 


5 11 
6.49 


13 


17.20 


5.14 
6.46 


17.37 


5.12 
6.48 


17.55 


5.10 
6.50 


18.13 


5.08 
6.52 


18.33 


5.07 
6.53 


14 


18.42 


5.10 
6.50 


19.00 


5.08 
6.52 


19.19 


5.06 
6.54 


19.39 


5.04 
6.56 


20.00 


5.02 
6.58 


15 


20.03 


5.06 
6.54 


20.23 


5.04 
6.56 


20.44 


5.02 

6.58 


21.05 


5.00 
7.00 


21.28 


4.58 
7.02 


16 


21.25 


5.02 
6.58 


21.46 


5.00 
7.00 


22.08 


4.53 

7.07 


22.32 


4.56 
7.04 


22.57 


4.53 

7.07 


17 


22.48 


4.58 
7.02 


23.10 


4.56 
7.04 


23.34 


4.54 

7.06 


23.59 


4.51 

7.09 


24.25 


4.49 
7.11 


18 


24.10 


4.54 
7.06 


21.34 


4.52 

7.08 


25.00 


4.49 
7.11 


25.26 


4.47 
7.13 


25.55 


4.44 
7.16 


19 


25.33 


4.50 
7.10 


25.59 


4.48 
7.12 


26.26 


4.45 

7.15 


26.55 


4.42 
7.18 


27.25 


4.49 
7.21 


20 


26.57 


4.46 
7.14 


27.24 


4.43 
7.17 


27.53 


4.41 
7.19 


28.23 


4.38 
7.22 


28.56 


4.35 
7.25 


21 


28.21 


4.42 

7.18 


28.50 


4.39 
7.21 


29.20 


4.26 
7.34 


29.53 


4.33 

7.27 


30.27 


4.30 
7.30 


22 


29.46 


4.28 
7.22 


30.16 


4.35 
7.25 


30.49 


4.31 
7.29 


31.23 


4.28 
7.32 


31.59 


4.25 
7.35 


23 


31.11 


4.33 

7.27 


31.43 


4 30 
7.30 


32.18 


4.27 
7.33 


32.54 


4.23 

7.37 


33.33 


4.20 

7.40 


23^^ 


31.51 


4.31 

7.29 


32.24 


4.28 
7.32 


32.59 


4.24 
7.36 


33.37 


4.21 
7.39 


34.16 


4.17 
7.43 



R. and S. are applied for Lat. and Dec. of the same name. 

124 



The Sun's Amplitude— Continued. 



With rate of change of Azimuth, frc 
Prime Vertical, and with th 


m Sunrise to 
e timj of his 


the time of h 
Rising and S 


is crossing the 
elting. 


Lat. 


46° 


47 


o 


48 


o 


49 


o 


50° 


Dec. 


Change 
of Azimuth 
l°in sH m- 


Chan ire 
of Azimuth 
1° in 5% m. 


Chaii};e 
of AziTiiuth 
l^in 53^ m. 


Cha 
of Az 
l°in 5 


as;e 
muth 
V^ m. 


Change 
of Azimuth 
1" in 5^^ m. 




AMP. 


y\. M. 


AMP. 


H. M. 


AMP. 


H. M. 


AMP. 


H. M. 


AMP. 


H. M. 


r 


1°.26' 


R5.56 
S6.04 


1°.28' 


R5.56 
S 6.04 


1°. 30' 


R5.56 
S6.04 


1°. 31' 


R5.55 

S0.05 


1°.33' 


R5.55 
S6.05 


2 


2.53 


5.52 
6.08 


2.56 


5.51 
6.09 


2.59 


5.51 
6.09 


3.03 


5.51 
6.09 


3.07 


5.50 
6.10 


3 


4.19 


5.48 
6.12 


4.24 


5.47 
6.13 


4.29 


5.47 
6.13 


4.35 


5.46 
6.14 


4.40 


5.46 
6.14 


4 


5.40 


5.43 
6.17 


5.52 


5.43 
6.17 


5.59 


5.42 
6.18 


6.06 


5.42 
6.18 


6.14 


5.41 

6.18 


5 


7.12 


5.39 
6.21 


7.21 


5.38 
6.22 


7.29 


5.38 
6.22 


7..38 


5.37 
6.^3 


7.48 


5.36 
6.24 


6 


8.39 


5.35 
6.25 


8.49 


5.34 
6.26 


8.59 


5.33 

6.27 


9.10 


5.32 
6.28 


9.22 


5.31 

6.29 


7 


10.06 


5.31 
6.29 


10.18 


5.30 
6.30 


10.30 


5.29 
6.31 


10.42 


5.28 
6.32 


10.56 


5.26 
6.34 


8 


11.33 


5.27 
6.33 


11.46 


5.25 
6.35 


12.00 


5.24 
6.36 


12.15 


5.23 
6.37 


12.30 


5.21 
6.39 


9 


13.01 


5.22 
6.38 


13.16 


5.21 
6.39 


13.31 


5.19 
6.41 


13.48 


5.18 
6.42 


14.05 


5.16 
6.44 


10 


14.29 


5.18 
6 42 


14.45 


5.16 
6-44 


15.02 


5.15 
0.45 


15.21 


5.13 

6.47 


15.40 


5.11 
6.49 


11 


15.57 


5.14 
6.46 


16 15 


5.12 

6.48 


16.34 


5.10 
6.50 


16.54 


5.08 
6.52 


17.16 


5.06 
6.54 


12 


17.25 


5.09 
6.51 


17.45 


5.07 
6.53 


18.06 


5.05 
6.55 


18.29 


5.03 
6.57 


18.52 


5.01 
6.59 


13 


18.54 


5.05 
6.55 


19.16 


5.03 
6.57 


19.39 


5.01 
6.59 


20.03 


4 58 
7.02 


20.29 


4.56 

7.04 


14 


20.23 


5.00 

7 00 


20.47 


4.58 
7.02 


21.12 


4.56 
7.04 


21.38 


4.53 

7.07 


22.07 


4.51 

7.09 


15 


21.53 


4.56 

7.04 


22.18 


4.53 

7.07 


22.45 


4.51 

7.09 


23.14 


4.48 

7.ia 


23.45 


4.46 
7.J4 


16 


23.23 


4.5L 

7.09 


23.50 


4.48 
7.12 


24.20 


4.46 
744 


24.51 


4.43 
7.17 


25.24 


4.40 

7.20 


17 


24.53 


4.56 
7.14 


25.23 


4.53 
7.17 


25.55 


4.41 
7.19 


26.28 


4.38 
7.22 


27.03 


4.35 

7.25 


18 


26.25 


4.41 
7.19 


26.57 


4.38 
7.22 


27.30 


4.35 
7.25 


28.06 


4.32 

7.28 


28.44 


4.29 
7.31 


19 


27.57 


4.36 
7.24 


28.31 


4.33 

7.27 


29.07 


4.30 

7 30 


29.45 


4.27 
7.33 


30.26 


4.23 
7.37 


2U 


29.30 


4.31 

7.29 


30.06 


4 28 
7.32 


30.44 


4.25 
7.35 


31.25 


4.21 

7.39 


32.09 


4.17 
7.43 


21 


31.03 


4.26 
7.34 


31.42 


4.23 
7.37 


32.23 


4.19 
7.41 


33.07 


4.15 
7.45 


33.53 


4.11 
7.49 


22 


32.38 


4.21 
7.39 


33.19 


4.17 
7.43 


34.03 


4.13 

7.47 


34.49 


4.09 
7.51 


35.39 


4.05 

7.55 


23 


34.14 


4.16 
7.44 


34.57 


4.12 

7.48 


35.44 


4.07 
7.53 


36.33 


4.03 

7.57 


37.26 


3.58 
8.02 


233^ 


34.59 


4.13 

7.47 


35.43 


4.09 
7.51 


36.31 


4.05 
7.55 


37.22 


4.00 
8.00 


38.17 


3.55 

8.05 



When Lat. and Dec. are unlike, R. and S. must change places with each other. 

125 



TABLE V. 




Sun's Declination for every 2nd day, with 


corresponding 


Equation of Time, for 1891. 






January. 


February. 


March. 


April. 


May. 


June. 






South. 


South. 


South. 


North. 


North. 


North. 




JNorih. 


>^ 


























>. 




Dec. 


Eq. 


Dec. 


Kq. 


Dec. 


Kq. 


Dec. 


Eq. 


Dec. 


Eq. 


Dec. 


Eq. 






S. 




S. 




S. 




N. 




N. 




N. 








o t 


m. 


o / 


m. 


o / 


m. 


o / 


m. 


o / 


m. 


c / 


m. 




1 


23.01 


— 4 


17.06 


—14 


7.35 


—13 


4.33 


—4 


15.05 


+3 


22.04 


+2 


1 


3 


22.50 


— 5 


16.31 


—14 


6.49 


—12 


5.19 


—3 


15 40 


+3 


22.19 


+2 


A 


5 


22.37 


— 6 


15.55 


—14 


6.03 


—12 


6.04 


-3 


16.15 


+4 


22.33 


^2 


5 


7 


22.23 


— 6 


15.18 


—14 


5.16 


—11 


6.50 


-2 


16.49 


+ 4 


22.46 


+ 1 


7 


9 


22.06 


— 7 


14.40 


—14 


4.29 


—11 


7.35 


—2 


17.21 


+4 


22.56 


+ 1 


9 


11 


21.48 


— 8 


14.01 


-14 


3.42 


—10 


8.19 


-1 


17.53 


+4 


23.06 


+1 


11 


13 


21.29 


— 9 


13.21 


—14 


2.55 


—10 


9.03 


—1 


18.23 


+4 


23.13 


+ 


13 


15 


21.07 


—10 


12.40 


—14 


2.08 


— 9 


9.46 


—0 


18.52 


+ 4 


23.19 


—0 


15 


17 


20.44 


—10 


11.58 


—14 


1.20 


— 9 


10.29 


+0 


19.20 


4-4 


23.24 


-1 


17 


19 
21 


20.20 
19.54 


—11 
—12 


11.16 
10.33 


—14 
—14 


S0.33 


— 8 
-7 


11.10 
11.52 


+ 1 
-fl 


19.46 
20.11 


+ 4 
+4 


23.26 
23.27 


—1 
—1 


19! 
21 


NO. 15 


23 


19.26 


— ]2 


9.49 


—14 


1.02 


— 7 


12.32 


+2 


20.35 


^4 


23.27 


—2 


23 


25 


18.57 


-13 


9.05 


-13 


1.49 


— 6 


13.11 


+2 


20.57 


+ 3 


23.24 


—2 


25 


27 


18.27 


—13 


8.20 


—13 


2 36 


— 6 


13.50 


+ 2 


21.18 


+ 3 


23.20 


—3 


27 i 


29 


17.55 


—13 


7.35 


-13 


3.23 


— 5 


14.28 


-f3 


21.37 


+3 


23.15 


—3 


29; 


31 


17.23 


—14 






4.09 


— 4 







21.55 


-r3 







31 


Note. — The sign prefixed to the Equ 


ation of Time, 


in the above table, is that for reducing n 


lean time to 


apparent time. 





126 



TABLE V.-Continued. 


Sun's Declination for every 2nd day, with corresponding 


Equation of Time, for 1891. 




July. 


August. 


September. 


October. 


November. 


December. 






North. 


North. 


North. 


South. 


South. 


South. 




South. 


>» 


























>. 




Dec. 


Eq. 


Dec. 


Eq. 


Dec. 


Eq. 


Dec. 


Eq. 


Dec. 


Eq. 


Dec. 


Eq. 






N. 




N. 




N. 




S. 




S. 




S. 








o / 


m. 


o / 


m. 


o / 


m. 


o r 


m. 


o / 


m. 


o / 


m. 




1 


23.08 


-4 


18.03 


—6 


8.18 


+ 


3.11 


+ 10 


14.26 


+16 


21.49 


+11 


1 


3 


22.59 


—4 


17.32 


-6 


7.34 


+ 1 


3.57 


+ 11 


15.04 


+16 


22.07 


+10 


3 


5 


22.48 


-4 


17.00 


—6 


6.50 


+ 1 


4.44 


+ 12 


15.41 


+16 


22.^3 


+ 9 


5 


7 


22.36 


-5 


16 27 


—6 


6.05 


+ 2 


5.30 


+ 12 


16.17 


+16 


22.37 


+ 8 


7 


9 


22.23 


—5 


15.53 


—5 


5.20 


+ 3 


6.16 


+ 13 


16.52 


+16 '22.50 


+ 8 


9 


11 


22.08 


—5 


15.18 


—5 


4.35 


+ 3 


7.01 


+ 13 


17.26 


+ 16 23.01 


+ 7 


11 


13 


21.51 


—5 


14.42 


- 5 


3.49 


+ 4 


7.46 


+ 14 


17.59 


+ 16 23.10 


+ 6 


13 


15 


21.33 


—6 


14.05 


—4 


3 03 


+ 5 


8.31 


+ 14 


18.30 


+ 15.23.17 


+ 5 


15 


17 


21.13 


-6 


13 27 


-4 


2.16 


+ 5 


9.15 


+ 15 


19.00 


+ 15 23.22 


+ 4 


17 


19 


20.52 


—6 


12.48 


—4 


1.30 


+ 6 


9.59 


+ 15 


19.29 


+ 15 23.25 


+ 3 


19 


21 
23 


20.30 
20.06 


-6 
-6 


12.08 
11.28 


-3 
-3 


N.43 


+ 7 
+ 7 


10.42 
11.24 


+ 15 
+ 16 


19.56 
20.21 


+ 14|23.27 
+ 14:23.27 


+ 2 

+ 1 


21 
23 


S. 4 


25 


19.41 


—6 


10.47 


—2 


.50 


+ 8 


12 06 


+16 


20.46 


+13 23.25 


— 


25 


27 


19.14 


—6 


10.05 


—2 


1.37 


+ 9 


12.47 


+ 16 


21.08 


+ 12 23.20 


— 1 


27 


29 


18.47 


—6 


9.23 


—1 


2.24 


+10 


13 27 


+16 


21.30 


+ 12 23.14 


— 2 


29 


31 


18.18 


—6 


8.40 


-0 






14.07 


+16 




23.07 


— 3 


31 


Note. — The sign prefixed to the Equation of Time, 


in the above table, is that for reducing mean time to 


apparent time. . 



127 



Azimuth and Hour Angle, for Latitude 






and Declination.— Table VI. 


LATITUDE 42^. 






^ZIiyCTJTBCS. 


jJeciiiiiiLJiuii. 


15° 


20° 


25° 


30^ 


35° 


40° 


45° 


50° 


55° 


60° 




H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H. M. 




+24° 


.21 


.28 


.35 


.43 


.52 


1.01 


1.11 


1.22 


1.35 


1.49 




22 


.22 


.30 


.29 


.47 


.57 


1.06 


1.17 


1.29 


1.43 


1.57 




20 


0.24 


0.33 


0.42 


0.51 


1.01 


1.12 


1.23 


1.30 


1.50 


2.06 


TJ 


18 


.26 


.35 


.45 


.55 


1.05 


1.17 


1.29 


1.42 


1.57 


2.13 


3 


16 


.28 


.37 


.48 


.58 


1.09 


1.21 


1.34 


1.48 


2.04 


2.21 


*rt 


14 


.29 


.40 


.50 


1.02 


1.13 


1.26 


1.40 


1.55 


2.11 


2.28 


-1 


\l 


.31 


.42 


.53 


1.05 


1.17 


1.31 


1.45 


2.00 


2.17 


2.36 


0) 


0.33 


0.44 


0.56 


J. 08 


1.21 


1.35 


1.50 


2.06 


2.24 


2.43 


^ 


8 


.34 


.46 


.59 


1.12 


1.25 


1.40 


1.55 


2.12 


2.30 


2.50 


'^ 


6 


.36 


.48 


1.01 


1.15 


1.29 


1.44 


2.00 


2.18 


2.36 


2.57 




4 


.38 


.51 


1.04 


1.18 


1.33 


1.49 


2.05 


2.23 


2.42 


3.03 




+ 2 


.39 


.53 


1.07 


1.21 


1.37 


1.53 


2.10 


2.29 


2.49 


3.10 





0.41 


0.55 


1.09 


1.24 


1.40 


1 57 


2.15 


2.34 


2.55 


3.17 




— 2 


.42 


.57 


1.12 


1.28 


1.44 


2.02 


2.20 


2.40 


3.01 


3.24 




4 


.44 


.59 


1.15 


1.31 


1.48 


2.06 


2.25 


2.45 


3.07 


3.30 


V 


6 


.45 


1.01 


1.17 


1.34 


1.52 


2.10 


2.30 


2.51 


3.13 


3.37 


3 


8 


.47 


1.03 


1.20 


1.37 


1.56 


2.15 


2.35 


2.57 


3.19 


3.44 




10 


0.49 


1.05 


1.23 


1.41 


1.59 


2.19 


2.40 


3.02 


3.26 


3.51 


^ 


12 


.50 


1.08 


1.25 


1.44 


2.03 


2.24 


2.45 


3.08 


3.32 


3.58 


14 


.52 


1.10 


1.28 


1.47 


2.07 


2.28 


2.51 


3.14 


3.39 


4.05 


J^ 


16 


.54 


1.12 


1.31 


1.51 


2.11 


2.33 


2.56 


3.20 


3.46 


4.13 


•-= 


18 


.55 


1.14 


1.34 


1.54 


2.16 


2.38 


3.01 


3.26 


3.53 


4.20 


^ 


20 
22 


0.57 
,59 


1.17 
1.19 


1.37 
1.40 


1.58 
2.02 


2.20 
2.24 


2.43 
2.48 


3.07 
3.13 


3.33 
3.39 


4.00 
4.07 


4.28 


4.36 




—24 


1.01 


1.22 


1.43 


2.06 


2.29 


2.53 


3.19 


3.46 


4.1^^ 


4.45 




















125° 


120° 



128 





Table VI - 


-Continued. 


LATITUDE 42^. 




.A.ZiX3y[:TJTSLS. 1 


Declination. 






■ 
























ea^' 


66° 


69° 


72° 


75° 


78° 


81° 


84° 


87° 


90° 




H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H.M. 


H. M. 


H.M. 


H. M. 


24° 


1 58 


2.08 


2.19 


2.3i 


2.42 


2.57 


3.12 


3.27 


3.44 


4.01 


22 


2.07 


2.18 


2.29 


2.41 


2.54 


3.08 


3.23 


3.39 


3.56 


4.13 


20 


2.16 


2.27 


2.39 


2.51 


3.05 


3.19 


3.34 


3 .50 


4.07 


4.25 


18 


2.24 


2 36 


2.48 


3.01 


3.14 


3.29 


3.44 


4.01 


4.18 


4.35 


16 


2.32 


2.44 


2.56 


3.10 


3.24 


3.39 


3.55 


4.11 


4.28 


4.46 


14 


2.40 


2 52 


3.05 


3.19 


3.33 


3.48 


4.04 


4.21 


4.38 


4.56 


12 


2.47 


3 00 


3.13 


3.27 


3.42 


3.57 


4.14 


4.30 


4 48 


5.05 


10 


2.55 


3.08 


3.22 


3.36 


3.51 


4.06 


4.23 


4.40 


4.57 


5.15 


8 


3.02 


3.16 


3.30 


3.44 


3.59 


4.15 


4.32 


4.49 


5.06 


5.24 


6 


3.10 


3.23 


3.37 


3.52 


4.08 


4.24 


4.41 


4.58 


5 15 


5.33 


4 


3.17 


3.31 


3.45 


4.00 


4.16 


4.33 


4.49 


5.07 


5.24 


5.42 


+ 2 


3.24 


3.38 


3.53 


4.08 


4.24 


4.41 


4.58 


5.16 


5.33 


5.51 





3.31 


3.45 


4.01 


4.16 


4.33 


4.50 


5.07 


5.24 


5.42 


6.00 




— 2 
4 
6 
8 
10 
12 
14 


3.38 
3.45 
3.52 
3.59 
4.07 
4 14 
4.22 


3.53 

4.00 
4.08 
4.15 
4.23 
4.31 
4.39 


4 08 
4.16 
4.24 
4.32 
4.40 
4 48 
4.56 


4.24 
4.32 
4.41 

4.49 
4.57 
5.05 


4.41 
4.49 
4.58 
5.06 
5.15 


4.58 
5.06 
5.15 
5.24 


5.15 
5.24 
5.33 


5.33 
5.42 


5.51 


6.09 
6.18 
6.27 
6.36 
6.45 
6.55 
7.04 


6.00 
6.09 
6.18 
6.27 
6.37 
6.46 


5.51 

6.00 
6.09 
6.18 
6.28 


5.42 
5.51 
6 00 
6.09 


5.33 
5.42 
5.51 


5.23 
5,32 


5.14 


16 

18 


4.30 
4.38 


4.47 


5.05 
5.14 


5.23 
5.32 


5.41 

5.51 


6.00 
6.10 


6.19 
6.29 


6.38 
6.48 


6.56 
7.06 


7.14 
7.25 


4.58 


20 


4.46 


5.04 


5.23 


5.42 


6.01 


6.20 


6.39 


6.58 


7.17 


7.35 


22 


4.55 


5.13 


5.32 


5.52 


6.11 


6 31 


6.50 


7.10 


7.28 


7.47 


24 


5.04 


5.23 


5. 42 


6.02 


6.23 


6.42 


7.02 


7.21 


7.40 


7.59 




117° 


114^ 


111° 


108° 


105° 


102° 


99° 


96° 


93° 


90° 



129 



Azimuth and Hour Angle, for Latitude 






and Declination —Table VI. 


LATITUDE 430. 






^zinynTJTiHiS- 




15° 


20° 


25° 


30^ 


35° 40° 


45° 


50° 


55° 


60° 




H.M. 


H. M. 


H. M. 


H. M. 


H. M. H. M. 


H. M, 


H. M. 


H. M. 


H. M. 




+24° 


.22 


.29 


.37 


.46 


.55 


1.04 


1.15 


1.26 1.39 


1.54 




22 


.24 


.32 


.40 


.49 


.59 


1.09 


1.21 


1.33 i 1.47 


2.02 




20 


0.25 


0.34 


0.43 


0.53 


1.03 


1.14 


1.26 


1.39 i 1.54 


2.10 


TJ 


18 


.27 


.37 


.46 


.57 


1.08 


1.19 


1.32 


1.40 1 2.01 


2.17 


3 


16 


.29 


.39 


.49 


1.00 


1.12 


1.24 


1.37 


1.52 


2.07 


2.23 


"5 


14 


.30 


.41 


.52 


1.03 


1.16 


1.29 


1.43 


1.58 


2.14 


2.32 


hJ 


12 


.32 


.43 


.55 


1.07 


1.20 


1.33 


1.48 


2.03 


2.20 


2.39 


0) 


10 


0.34 


0.45 


0.57 


1.10 


1.23 


1.38 


1.52 


2.09 


2.27 


2.46 


.id 


8 


.35 


.47 


1.00 


1.13 


1.27 


1.42 


1.58 


2.15 


2.33 


2.52 


'h^ 


6 


.37 


.50 


1.03 


1.17 


1.31 


1.46 


2.03 


2.20 


2.39 


2.59 




4 


.38 


.52 


1.05 


1.20 


1.35 


1.51 


2.07 


2.26 


2.45 


3.06 




+ 2 


.40 


.54 


1.08 


1.23 


1.38 


1.55 


2.12 


2.31 


2.51 


3.12 





0.41 


0.56 


1.11 


1.26 


1.42 


1,59 


2.17 


2.36 


2.57 


3.19 




— 2 


.43 


.58 


1.13 


1.29 


1.46 


2.03 


2.22 


2.42 


3.03 


3.26 




4 


.45 


1.00 


1.16 


1.32 


1.50 


2.08 


2.27 


2.47 3.09 


3.32 


u 


6 


.46 


1.02 


1.18 


1.35 


1.53 


2.12 


2.32 


2.53 


3.15 


3.39 


3 


8 


.48 


1.04 


1.21 


1.39 


1.57 


2.16 


2.37 


2.58 


3.21 


3.46 




10 


0.49 


1.06 


1.24 


1.42 


2.01 


2.21 


2.42 


3.04 


3.27 


3.52 




12 


.51 


1.08 


1.26 


1.45 


2.05 


2.25 


2.46 


3.09 


3.34 


3.59 


14 


.52 


1.11 


1.29 


1.48 


2.09 


2.30 


2.52 


3.15 


3.40 


4.06 


^ 


16 


.54 


1.13 


1.32 


1.52 


2.13 


2.34 


2.57 


3.21 


3.47 


4.13 


— 


18 


.56 


1.15 


1.35 


1.55 


2.17 


2.39 


3.03 


3.27 


3.53 


4.21 


:::> 


20 
22 


0.58 
.59 


1.17 
1.20 


1.38 
1.41 


1.59 
2.03 


2.21 
2.25 


2.44 
2.49 


3. 08 
3.14 


3.33 
3.40 


4.00 
4.07 


4.28 


4.36 




—24 


1.01 


1.22 


1.44 


2.06 


2.30 


2.54 


3.20 


3.47 


4.1'> 


4.44 


1 

























130 



Table VI.— Continued. 


LATITUDE 43''. 




^ZXl^TJTHIS. 


Declination. 


























63° 


66° 


69° 


72° 


75° 


78° 


81° 


84° 


87° 


90° 




H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H.M. 


H. M. 


+24° 


2 03 


2.13 


2.24 


2.36 


2.49 


3.02 


3.17 


3.32 


3.49 


4.06 


22 


2.12 


2.22 


2.34 


2.46 


2.59 


3.13 


3.28 


3.44 


4.00 


4.17 


20 


2.20 


2.31 


2.43 


2.56 


3.09 


3.23 


3.38 


3 54 


4.11 


4.28 


18 


2.28 


2 40 


2.52 


3.05 


3.19 


3.33 


3.48 


4.04 


4.21 


4.38 


16 


2.36 


2.48 


3.00 


3.14 


3.28 


3.43 


3.58 


4.14 


4.31 


4.48 


14 


2.43 


2 56 


3.09 


3.22 


3.37 


3.52 


4.07 


4.24 


4.41 


4.58 


IG 


2.51 


3 03 


3.17 


3.31 


3.45 


4.00 


4.16 


4.33 


4 50 


5.07 


10 


2.58 


3.11 


3.25 


3.39 


3.54 


4.09 


4.25 


4.42 


4.59 


5.16 


8 


3.05 


3.18 


3.32 


3.47 


4.02 


4.18 


4.34 


4.51 


5.08 


5.25 


6 


3.12 


3.26 


3.40 


3.55 


4.]0 


4.26 


4.42 


4.59 


5 17 


5.34 


4 


3.19 


3.33 


3.48 


4.03 


4.18 


4.34 


4.51 


5.08 


5.25 


5.43 


+ 2 


3.26 


3.40 


3.55 


4.10 


4.26 


4.43 


4.59 


5.16 


5.34 


5.51 





3.33 


3.47 


4.03 


4.18 


4.34 


4.51 


5.08 


5.25 


5.42 


6.00 




— 2 
4 


3.40 
3.47 


3.55 
4.02 


4 10 
4.17 


4.26 
4.34 


4.42 
4.50 


4.59 
5.07 


5.16 
5.24 


5.33 
5.42 


5.51 


6.09 
6.17 


6.00 


6 

8 

10 

12 


3.54 
4.01 
4.08 
4.15 


4.09 
4.16 
4.24 
4.31 


4.25 
4.33 
4.40 
4.48 


4.41 
4.49 
4.57 
5.06 


4.58 
5.06 
5.15 


5.15 
5.24 


5.33 


5.51 
5.59 
6.08 
6.17 


6.08 
6.17 
6.26 
6.35 


6.26 
6.35 
6.44 
6.53 


5.41 
5.50 
5.59 


5.32 
5.41 


5.23 


14 


4.22 


4.39 


4.56 


5.14 


5.32 


5.50 


6.08 


6.26 


6.44 


7.02 


16 
18 


4.30 
4.38 


4.47 


5.05 
5.13 


5.23 
5.31 


5.41 
5.50 


5.59 
6.08 


6.17 
6.27 


6.36 
6.46 


6.54 
7.04 


7.12 

7.22 


4.55 
























20 


4.46 


5.04 


5.22 


5.41 


5.59 


6.18 


6.34 


6.56 


7.14 


7.32 


22 


4.54 


5.13 


5.31 


5.50 


6.09 6.28 


6.48 


7.06 


7.25 


7 43 


—24 


5.03 


5.22 


5.41 6.00 


6.20 6.39 


6.59 


7.1§ 


7.36 


7.54 




117° 


114° 


111° 


108° 


105° 102° 


99° 


96° 


93° 


90° 



131 



Azimuth and Hour Angle, for Latitude 




and Dec ination.— Tab e Vl. 


LATITUDE 44^. 


Declination. 


j^zi:]ynTJTH:s_ 


15° 


20^ 


2C° 33^ 1 35° 40° 


45° 


50° 


55° 


60° 




H. M. 


H. M. 


H. M. 


H . M . 


H . M . H . M . 


H. M. 


H. M. j H. M. 


H. M. 


-f 24° 


.23 


.31 


.39 


.48 


.57 1.07 


1.18 


1.30 ! 1.43 


1.58 


22 


.24 


.33 


.42 


.52 


1 02 , 1.12 


1 24 


1.36 


1.51 


2.06 


^ 20 


0.26 


0.36 


0.45 


0.55 


1.06 


1.17 


1.29 


1 43 


1.58 


2.14 


-o 18 


.28 


.38 


.48 


.59 


1.10 


1 22 ( 1 35 


1.49 


2.04 


2.21 


B 16 


.30 


.40 


.51 


1.02 


1.14 


1 27 


1.40 


1.55 


2.11 


2.28 


rt 14 


.31 


.42 


.54 


1.05 


1.18 


1.31 


1.45 


2.00 


2.17 


2.35 


2 12 


.33 


.44 


.56 


1.08 


1.22 ' 1 35 


1.50 


2.06 


2.23 


2.42 


a; 10 


0.35 


0.46 


0.59 


1.12 


1.25 1.40 


1 55 


2.12 


2.30 


2.49 


-. 8 


.36 


.49 


1.02 


1.15 


1.29 t 1.44 


2.00 


2.17 


2.36 


2.55 


6 


.38 


.51 


1.04 


1.18 


1.33 1 1.48 


2 05 


2.22 


2.42 


3.02 


4 


.39 


.53 


1.07 


1.21 


1.36 1 1.53 


2.10 


2.28 


2.47 


3.08 


+ 2 


.41 


.55 


1.09 


1.24 


1.40 j 1.57 


2.14 


2.33 


2.53 


3.15 





0.42 


0.57 


1.12 


1.27 


1 
1.44 1 2 01 

1 


2.19 


2.38 


2.59 


3.21 


— 2 


.44 


.59 


1.14 


1.31 


1.47 


2.05 


2 24 


2.44 


3.05 


3.27 


4 


.45 


1.01 


117 


1.34 


1.51 


2.09 


2.29 


2.49 


3.11 


3.34 


i? 6 


.47 


1.03 


1.19 


1.37 


1.55 


2.14 


2.33 


2-54 


3.17 


3.40 


5 8 


.48 


1.05 


1.22 


1.40 


1.58 


2.18 


2.38 


3.00 


3.23 


3 47 


•S 10 


0.50 


1.07 


1.25 


1.43 


2.02 


2 . 22 


2.43 


3 05 


3.29 


3.53 


^ 12 


.51 


1.09 


1.27 


1.46 


2.06 


2.26 


2.48 


3 11 


3.35 


4.00 


Z 14 


.53 


1.11 


1,30 


1.49 


2.10 


2.31 


2 53 


3.16 


3.41 


4.07 


^ 16 


.55 


1.13 


1.33 


1.53 


2.14 


2.35 


2.58 


3.22 


3.47 


4.14 


■■H 18 


.56 


1.16 


1.36 


1.56 


2.18 


2.40 


3.03 


3.28 


3.54 


4.21 


;d 20 
22 


0.58 
1.00 


1.18 
1.20 


1.38 
1.41 


2.00 
2 03 


2.22 
2 26 


2.45 
2.50 


3.09 
3.14 


3.34 


4.01 


4 28 


3.40 4.08 


4.36 


—24 


1.02 


1 . 23 [ 1 . 44 


2.07 


2.30 


2.55 


3.20 


3.47 


4.1=> 


4.44 










i 










125° 


120° 



132 





Table VL- 


-Continued. 




LATITUDE 


440, 


Declination. 


.i^ZinVCTJTS-S- 


















! 




63^ 


66° 


69° 


72° 


75° 


78° 


81° 


84° 


87° 


90° 




H. !\1. 


H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H . .M . 


H. M. 


H. M. 


H. M. 


+ 21^ 


2 08 2.18 


2.29 


2.41 


2.54 


3.07 


3.22 


3.37 


3.53 


4.10 


22 


2.16 1 2.27 


2.38 


2.51 


3.04 


3.18 


3.32 


3.48 


4.04 


4.21 


20 


2.24 


2.35 


2.47 


3.00 


3.13 


3.27 


3.42 


3 58 


4.14 


4.31 


18 


2 32 


2 44 


2.56 


3.09 


3.22 


3.37 


3.52 


4.08 


4.24 


4.41 


16 


2.40 


2.51 


3.04 


3.17 


3.31 


3.46 


4.01 


4.17 


4.34 


4.51 


14 


2.47 


2 59 


3.12 


3 26 


3.40 


3.55 


4.10 


4.26 


4.43 


5.00 


12 


2.54 


3 07 


3.20 


3.34 


3 48 


4.03 


4.19 


4.35 


4 52 


5.09 


10 


3.01 


3.14 


3.28 


3 42 


3.56 


4.12 


4.28 


4.44 


5.01 


5.18 


8 


3.08 


3.21 


3.35 


3.49 


4.04 


4.20 


4.36 


4.52 


5.09 


5.26 


6 


3.15 


3.28 


3.42 


3.57 


4.]2 


4.28 


4.44 


5.01 


5 18 


5.35 


4 


3.22 


3.35 


3.50 


4.05 


4.20 


4.36 


4.52 


5.09 


5.26 


5.43 


+ 2 


3.28 


3.42 


3.57 


4.12 


4.28 


4.44 


5.01 


5.17 


5.34 


5.52 





3.35 


3.49 


4.04 


4.20 


4.36 


4.52 


5.09 


5.26 


5.43 


6.00 




— 2 
4 


3.42 

3.48 


3.56 \ 4 12 


4.27 
4.35 


4.43 
4.51 


5.00 
5.08 


5.17 
5.25 


5.34 
5.42 


5.51 


6.08 
6.17 


4.03 


4.19 


5 , 59 


6 


3.55 


4.10 


4.26 


4.42 


4.59 


5.16 


5.33 


5.50 


6.08 


6.25 


8 
10 
12 
14 


4.02 
4.09 
4.16 
4.23 


4.18 
4.25 
4.32 
4.40 


4.34 
4.41 
4.49 
4.57 


4.50 
4.58 
5.06 


5.07 
5.15 


5.24 


5.41 
5.50 
5.58 
6.07 


5.59 
6.07 
6.16 
6.25 


6.16 
6.25 
6.33 
6.42 


6.34 
6.42 
6.51 

7.00 


5 32 
5.41 
5.49 


5.23 
5,31 


5.14 


16 
18 


4.30 
4.38 


4.47 


5.05 
5.13 


5.22 
5.31 


5.40 
5 49 


5.58 
6.07 


6.16 
6.25 


6.34 
6.43 


6.52 
7.01 


7.09 
7.19 


4.55 


20 


4.46 


5.03 


5.21 


5.40 


5.58 


6.I0 


6.35 


6.53 


7.11 


7.29 


22 


4.54 


5.12 


5.30 


5.49 


6.08 


6 26 


6.45 


7.03 


7.21 


7.39 


—24 


5.02 


5.21 


5 40 


5.58 


6.18 


6.37 


6.55 


7.14 


7.32 


7.50 




117° 


lir 


111° 


108° 


105° 


102° 


99° 


96° 


93° 


90° 



133 



Azimuth and Hour Angle, for Latitude 




and Dec ination— Table VL 


LATITUDE 45^. 


Declination. 


^ZXnVCTJTHIS. 


15° 


20° 


25° 


30^ 


35° 


40° 


45° 


50° 


55° 


60° 




H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


H.M. 


H. M. 


H. M. 


H. M. 


H. M. 


+24° 


.24 


.32 


.41 


.50 


1.00 


1.10 


1.21 


1.34 


1.47 


2.02 


22 


.26 


.35 


.44 


.54 


1.04 


1.15 


1.27 


1.40 


1.54 


2.10 


«5 20 


0.27 


0.37 


0.47 


0.57 


1.08 


1.20 


1.33 


1.46 


2.01 


2.18 


-S 18 


.29 


.39 


.50 


1.01 


1.12 


1.25 


1.38 


1.52 


2.08 


2.25 


B 16 


.31 


.41 


.52 


1.04 


1.16 


1.29 


1.43 


1.58 


2.14 


2.32 


rt 14 


.32 


.43 


.55 


1.07 


1.20 


1.33 


1.48 


2.03 


2.20 


2.39 


2 12 


.34 


.46 


.58 


1.10 


1.24 


1.38 


1.53 


2.09 


2.26 


2.45 


V 10 


0.35 


0.48 


1.00 


1.14 


1.27 


1.42 


1.58 


2.14 


2.32 


2.52 


^ 8 


.37 


.50 


1.03 


1.17 


1.31 


1.46 


2.02 


2 20 


2.38 


2.58 


^ 6 


.38 


.52 


1.05 


1.20 


1.35 


1.50 


2.07 


2.25 


2.44 


3.04 


4 


.40 


.54 


1.08 


1.23 


1.38 


1.55 


2.12 


2.30 


2.50 


3.11 


+ 2 


.41 


.56 


1.10 


1.26 


1.42 


1.59 


2.16 


2.35 


2.55 


3.17 





0.43 


0.58 


1.13 


1.29 


1.45 


2.03 


2.21 


2.40 


3.01 


3.23 


— 2 


.44 


1.00 


1.16 


1.32 


1.49 


2.07 


2 26 


2.46 


3.07 


3.29 


4 


.46 


1.02 


1.18 


1.35 


1.52 


2.11 


2.30 


2.51 


3.13 


3.35 


^ 6 


.47 


1.04 


1.21 


1.38 


1.56 


2.15 


2.35 


2.56 


3.18 


3.42 


3 8 


.49 


1.06 


1.23 


1 41 


2.00 


2.19 


2.40 


3.01 


3.24 


3.48 


•S 10 


0.50 


1.08 


1.26 


1.44 


2.03 


2.23 


2.44 


3.07 


3.30 


3.54 


i 11 


.52 


1.10 


1.28 


1.47 


2.07 


2.28 


2.49 


3.12 


3.36 


4.01 


.54 


1.12 


1.31 


1.50 


2.11 


2.32 


2 54 


3.17 


3.42 


4.08 


^ 16 


.55 


1.14 


1.34 


1.54 


2.15 


2.36 


2.59 


3.23 


3.48 


4.14 


"H 18 


.57 


1.16 


1.36 


1.57 


2.19 


2.41 


3.04 


3.29 


3.55 


4.21 


^ 20 

22 

—24 


.58 
1.00 
1.02 


1.19 
1.21 
1.23 


1.39 
1.42 
1.45 


2.00 
2.04 
2.08 


2.23 
2.27 
2.31 


2.46 
2.50 
2.55 


3.10 
3.15 
3.21 


3.35 
3.41 
3.47 


4.01 
4.08 


4.29 


4.36 
4.44 


4.15 




















125° 


120° 



134 



Table VI.— Continued. 


LATITUDE 45^. 




^ZIDVCTJTSIS. 


Declination. 
























63" 


66° 


69° 


72° 


75° 


78° 


81° 


84° 


87° 


90° 




H. M. 


H. M. 


H. M. 


H. M. 


H.M. 


H.M. 


H. M. 


H. M. 


H. M. 


H. M. 


+24° 


2.12 


2.23 


2.34 


2.46 


2.59 


3.12 


3.26 


3,42 


3.58 


4.14 


22 


2.20 


2.31 


2.43 


2.55 


3.08 


3.22 


3.37 


3.52 


4.08 


4.25 


20 


2.28 


2.39 


2.51 


3.04 


3.17 


3.31 


3.46 


4.02 


4.18 


4.35 


18 


2 36 


2.47 


3,00 


3.13 


3.26 


3.41 


3.56 


4.11 


4.27 


4.44 


16 


2.43 


2.55 


3.08 


3.21 


3.35 


3.49 


4.05 


4.20 


4.37 


4.53 


14 


2.50 


3 02 


3.15 


3 29 


3.43 


3.58 


4.13 


4.29 


4.45 


5.02 


12 


2.57 


3 10 


3.23 


3.37 


3.51 


4.06 


4.22 


4.38 


4 54 


5.11 


10 


3.04 


3.17 


3.30 


3.44 


3.59 


4.14 


4.30 


4.46 


5.03 


5.19 


8 


3.11 


3.24 


3.38 


3.52 


4.07 


4.22 


4.38 


4.54 


5.11 


5.28 


6 


3.17 


3.31 


3.45 


3.59 


4.34 


4.30 


4.46 


5.02 


5.19 


5.36 


4 


3.24 


3.38 


3.52 


4.07 


4.22 


4.38 


4.54 


5.10 


5.27 


5.44 


+ 2 


3.30 


3.44 


3.59 


4.14 


4.30 


4.45 


5.02 


5.18 


5.35 


5.52 





3.37 


3.52 


4.06 


4.21 


4.37 


4.53 


5.09 


5.26 


5.43 


6.00 




— 2 

4 

6 

8 

10 

12 

14 

16 

18 


3.43 
3.50 
3.56 
4.03 
4.10 
4.17 
4.24 
4.31 
4.38 


3.58 
4.05 
4.12 
4.19 
4.26 
4.33 
4.40 
4.47 


4.13 
4.20 

4.27 
4.34 
4.42 
4. 49 
4.57 


4.29 
4.36 
4.43 
4.51 
4.58 
5.06 


4.44 
4.52 
5.00 
5.07 
5.15 


5.01 
5.08 
5.16 
5.24 


5.17 
5.25 
5.33 


5.34 
5.42 


5.51 


6.08 
6.16 
6.24 
6.32 
6.41 
6.49 
6.58 
7.07 
7.16 


5.59 
6.07 
6.15 
6.24 
6.32 
6.41 
6.50 
6.59 


5.50 
5.58 
6.06 
6.15 
6.23 
6.32 
6.41 


5.41 
5.49 
5.57 
6.06 
6.14 
6.23 


5.32 
5.40 
5.48 
5.57 
6.06 


5.23 
5.31 
5.39 

5.48 


5.14 
5.22 
5.30 


5.04 
5.12 


4.55 


20 


4.46 


5.03 


5.21 


5.39 


5.57 


6.15 


6.33 


6.51 


7.08 


7.25 


22 


4.53 


5.11 


5.29 


5.47 


6.06 


6 24 


6.42 


7.00 


7.18 


7.35 


—24 


5.02 


5.20 


5. 38 


5.57 


6.15 


6.34 


6.53 


7.11 


7.28 


7.46 




117° 


114° 


111° 


108° 


105° 


102° 


99° 


96° 


93° 


90° 



135 



Azimuth and Hour Angle, for Latitude 




and Declination— Table VI. 


LATITUDE 46"^. 


Declination. 


-A^ZZDVCTTTIHIS- 


15° 


20° 


25° 


30^ 


35° 40° 


45° 


50° 


55° 


60° 




H. M. 


H. M. 


H. M. 


H. M. 


1 

H. M. j H. M. 


H. M. 


H. M. 


H. M. 


H. M. 


+24° 


.25 


.34 


.43 


.52 


1.02 


1.13 


1.25 


1.38 


1.51 


2.07 


22 


.27 


.36 


.46 


.56 


1.07 


1.18 


1.30 


1.44 


1.58 


2.14 


.\ 20 


0.28 


0.38 


0.49 


0.59 


1.11 


1.23 


1.36 


1.50 


2.05 


2.21 


-S 18 


.30 


.40 


.51 


1.03 


1.14 


1.27 


1.41 


1.56 


2.11 


2.28 


2 16 


.32 


.43 


.54 


1.06 


1.18 


1.32 


1.46 2.01 


2.17 


2.35 


1i 14 


.33 


.45 


.57 


1.09 


1.22 


1.36 


1.51 


2.07 


2.23 


2.42 


^ 12 


.35 


.47 


.59 


1.12 


1.26 


1.40 


1.55 


2.12 


2.29 


2.48 


0) 10 


0.36 


0.49 


1.02 


1.15 


1.29 


1.41 


2.00 


2.17 


2.35 


2.54 


^ 8 


.38 


.51 


1.04 


1.18 


1.33 


1.48 


2.05 


2 22 


2.41 


3.01 


^ 6 


.39 


.53 


1.07 


1.21 


1.36 


1.52 


2.09 


2.28 


2.46 


3.08 


4 


.41 


.55 


1.09 


1.24 


1.40 


1.56 


2.14 


2.33 


2.52 


3.13 


+ 2 


.42 


.57 


1.12 


1.27 


1.43 


2.00 


2.18 


2.38 


2.58 


3.19 





0.44 


0.59 


1.14 


1.30 


1.47 2.04 


2.23 


2.43 


3.03 


3.25 


— 2 


.45 


1.01 


1.17 


1.33 


1.50 2.03 


2.27 


2.48 


3.09 


3.31 


4 


.47 


1.03 


1.19 


1.36 


1.54 I 2.12 


2.32 


2.53 


3.14 


3.37 


V 6 


.48 


1.05 


1.22 


1.39 


1.57 


2.16 


2.37 


2.58 


3.20 


3.42 


^ 8 


.50 


1.07 


1.24 


1.42 


2.01 


2.21 


2.41 


3.03 


3.25 


3.49 


•S 10 


0.51 


1.09 


1.27 


1.45 


2.05 


2.25 


2.46 


3.08 


3.31 


3.56 


5 12 
^ 14 


.53 


1.11 


1.29 


1.48 


2.08 


2.29 


2.50 


3.13 


3.37 


4.02 


.54 


1.13 


1.32 


1.51 


2.12 


2.33 


2. 55 


3.19 


3.43' 


4.08 


J 16 


.56 


1.15 


1.34 


1.55 


2.16 


2.37 


3.00 


3.24 


3.49 


4.15 


•-§ 18 


.57 


1,17 


1.37 


1.58 


2.19 


2.42 


3.05 


3.30 


3.55 


4.22 


^ 20 

22 

—24 


0.59 
1.01 
1.02 


1.19 
1.21 
1.24 


1.40 
1.43 
1.46 


2.01 
2.04 
2.08 


2.23 
2.27 
2.32 


2.46 
2.51 
2.56 


3.10 
3.16 
3.21 


3.36 
3.42 
3.48 


4.01 
4.08 ' 


4.29 


4.36 
4.43 


4.15 




















125° 


120° 



136 



Table VI —Continued. 


LATITUDE 46^. 




^ZZnSdlTJTHIS. 


Declination. 


























63° 


66° 


69° 


72° 


75° 


78° 


81° 


84° 


87° 


90° 




H. M. 


H. M. 


H.M. 


H. M. 


H.M. 


H.M. 


H. M. 


H. M. 


H. M. 


H. M . 


+ 24° 


2 16 


2.27 


2.38 


2.50 


3.03 


3.17 


3.31 


3.46 


4.02 


4 18 


22 


2.25 


2.35 


2.47 


3.00 


3.12 


3.27 


3.41 


3.56 


4.12 


4.28 


20 


2.32 


2.43 


2.55 


3.08 


3.21 


3.36 


3.50 


4 05 


4.21 


4.38 


18 


2 39 


2 51 


3.03 


3.16 


3.30 


3.44 


3.59 


4.14 


4.30 


4.47 


16 


2.47 


2.58 


3.11 


3.24 


3.38 


3.53 


4.08 


4.23 


4.39 


4. 56 


14 


2.53 


3 06 


3. 19 


3.32 


3.46 


4.01 


4.16 


4.32 


4.48 


5 04 


12 


3.00 


3 13 


3.26 


3.40 


3 54 


4.09 


4.24 


4.40 


4 56 


5.13 


10 


3.07 


3.20 


3.33 


3.47 


4.02 


4.17 


4.32 


4.48 


5. 04 


5.21 


8 


3.13 


3.26 


3.40 


3.54 


4.09 


4.25 


4.40 


4.56 


5.1-2 


5.29 


6 


3.20 


3.33 


3.47 


4.02 


4.]6 


4.32 


4.48 


5.04 


5 20 


5.37 


4 


3.26 


3.40 


3.54 


4.09 


4.24 


4.40 


4.55 


5.11 


5.28 


5.45 


+ 2 


3.32 


3.46 


4.01 


4.16 


4.31 


4.47 


5.03 


5.19 


5.36 


5.52 





3.39 


3.53 


4.03 


4.23 


4.38 


4.55 


5.10 


5.27 


5.43 


6.00 




— 2 
4 


3.45 
3.51 


4.00 
4 06 


4 14 
4.21 


4.30 
4.37 


4.46 
4.53 


5.02 
5.09 


5.18 
5.25 


5.34 
5.42 


5.51 


6.08 
6.15 


5.59 


6 
8 
10 
12 
14 
16 


3.58 
4.04 
4.11 
4.17 
4.24 
4.31 


4.13 
4.19 
4.26 
4.33 
4.40 
4.47 


4.28 
4.35 
4.42 
4.49 
4.57 


4.44 
4.51 

4.58 
5.06 


5.00 
5.08 
5.15 


5.17 
5.24 


5.33 


5.50 
5.58 
6.06 
6.14 
6.22 
6.30 


6.07 
6.14 
6.22 
6.30 
6.39 
6.47 


6.23 
6.31 
6.39 
6.47 
6 56 
7.04 


5.41 
5.49 
5.57 
6.05 
6.13 


5.32 
5.40 
5.48 
5.56 


5.23 
5.30 
5.38 


5.13 
5.21 


5.04 


18 


4,38 


4.55 


5.12 


5.29 


5.47 


6.05 


6.22 


6.39 


6.56 


7.13 


20 


4.45 


5.03 


5.20 


5.37 


5.55 


6.13 


6.31 


6.48 


7.05 


7 22 


22 


4.53 


5.10 


5.28 


5.46 


6.04 


6 22 


6.40 


6.58 


7.15 


7.32 


-24 


5.01 


5.19 


5.37 


5.55 


6.13 


6.32 


6.50 


7.07 


7.25 


7.42 




117° 


lir 


111° 


108° 


105° 


102° 


99° 96° 


93° 


90° 



137 



Azimuth and Hour Angle, for Latitude 




and Declination— Table VI. 


LATITUDE 47^. 


Declination. 


.A^ZIDyCTTTHES. 
























15° 


20° 


25° 


30^ 


35^ 


40° 


45° 


50° 


55° 


60° 




H. M. 


H. M. 


H.M. 


H.M. 


H, M. 


H. M. 


H. M. 


H. M. 


H.M. 


H. M. 


+24° 


.26 


.35 


.45 


.54 


1.05 


1.16 


1.28 


1.41 


1.55 


2.11 


22 


.28 


.37 


.47 


.58 


1 09 


1.21 


1.33 


1.47 


2.02 


2.18 


.; 20 


0.29 


0.40 


0.50 


1.01 


1.13 


1.25 


1.38 


1.53 


2.08 


2.25 


-S 18 


.31 


.42 


.53 


1.04 


1.17 


1.30 


1.44 


1.58 


2.14 


2.32 


2 16 


.33 


.44 


.56 


1.08 


1.20 


1 34 


1.48 


2.04 


2.20 


2.38 


'5 14 


.34 


.46 


.58 


1.11 


1.24 


1.38 


1.53 


2.09 


2.26 


2.45 


2 12 


.36 


.48 


1.01 


1.14 


1.28 


1.42 


1.58 


2.14 


2.32 


2.51 


0) 10 


0.37 


0.50 


1.03 


1.17 


1.31 


1.46 


2.02 


2.19 


2.38 


2.57 


•i 8 


.38 


.52 


1.06 


1.20 


1.35 


1.50 


2.07 


2 25 


2.43 


3.03 


^ 6 


.40 


.54 


1.08 


1.23 


1.38 


1.54 


2.11 


2 30 


2.49 


3.09 


4 


.41 


.56 


1.11 


1.26 


1.42 


1 58 


2.16 


2.34 


2.54 


3.15 


+ 2 


.43 


.58 


1.13 


1.29 


1.45 


2.02 


2.20 


2.39 


3.00 


3.21 





0.44 


1.00 


1.15 


1.32 


1.48 


2.06 


2.25 


2.44 


3.05 


3.27 


— 2 


.46 


1.02 


1.18 


1 34 


1.52 


2.10 


2.29 


2.49 


3.10 


3.33 


4 


.47 


1.03 


1.20 


1.37 


1.55 


2.11 


2.34 


2.54 


3.16 


3.39 


ii 6 


.49 


1.06 


1.23 


1.40 


1.59 


2.18 


2.38 


2.59 


3.21 


3.44 


1 8 


.50 


1.07 


1.25 


1.43 


2.02 


2.22 


2.42 


3.04 


3.27 


3.50 


•2 10 


0.52 


1.09 


1.27 


1.46 


2.06 


2.26 


2.47 


3 09 


3.32 


3.57 


^ 12 
"^ 14 


.53 


1.11 


1.30 


1.49 


2.09 


2.30 


2.52 


3 14 


3.38 


4.03 


.55 


1.13 


1.83 


1.52 


2.13 


2.34 


2 56 


3.19 


3.44 


4.09 


i^ 16 


.56 


1.15 


1.35 


1.55 


2.16 


2.38 


3.01 


3.25 


3.50 


4.15 


'■^ 18 
l:^ 20 


.58 
0.59 


1.18 
1.20 


1.38 
1.40 


1.59 
2.02 


2.20 
2.24 


2.43 
2.47 


3.06 
3.11 


3.30 
3.3d 


3.56 
4.02 


4.22 


4.29 


22 
—24 


1.01 
1.03 


1.22 
1.24 


1.43 
1.46 


2.05 
2.09 


2 28 
2.32 


2.52 
2.56 


3.16 
3.31 


3.42 

3.48 


4.08 


4.36 
4.43 


4.15 




















125° 


120° 



138 



Tabe VI— Continued. 


LATITUDE 47^. 


Declination. 


.A-ZinVwdlTTTSIS- 
























63" 


66° 


69° 


72° 


75° 


78° 


81° 


84° 


87° 


90° 




H.M. 


H. M. 


H. M. 


H.M. 


H.M. 


H.M. 


H.M. 


H. M. 


H. M. 


H, M. 


+24° 


2 21 


2.32 


2.43 


2.55 


3.08 


3.21 


3.35 


3,50 


4.06 


4.22 


22 


2.29 


2.40 


2.51 


3.04 


3.17 


3.30 


3.45 


4.00 


4.15 


4.31 


20 


2.86 


2.47 


2.59 


3.12 


3.25 


3.39 


3.54 


4 09 


4.24 


4.41 


18 


2 43 


2 55 


3.07 


3.20 


3.33 


3.48 


4.02 


4.17 


4.83 


4.49 


16 


2.50 


3.02 


3.14 


3.28 


3.41 


3.56 


4 10 


4.26 


4.42 


4.58 


14 


2.57 


3 09 


3.22 


3.35 


3.49 


4.04 


4.18 


4.84 


4.50 


5.06 


12 


3.03 


3.16 


3.29 


3,42 


3 57 


4.11 


4.26 


4.42 


4 58 


5.14 


10 


3.10 


3.22 


3.36 


3.49 


4.04 


4.19 


4.34 


4.50 


5.06 


5.22 


8 


3.16 


3.29 


3.43 


3.57 


4.11 


4.26 


4.42 


4.57 


5.14 


5.30 


6 


3.22 


3.35 


3.49 


4.04 


4.]8 


4.84 


4.49 


5.05 


5 21 


5.37 


4 


3.28 


3.42 


3.56 


4.11 


4.25 


4.41 


4.56 


5.12 


5.29 


5.45 


+ 2 


3.34 


3.48 


4.03 


4.17 


4.33 


4.48 


5.04 


5.20 


5.36 


5.53 





3.41 


3.55 


4.09 


4.24 


4.40 


4.55 


5.11 


5.27 


5.44 


6.00 




— 2 
4 


3.47 
3.53 


4.01 

4.07 


4 16 
4.22 


4.31 

4.38 


4.47 
4.54 


5.02 
5.10 


5.18 
5.26 


5.35 
5.42 


5.51 


6.07 
6.15 


5.59 


6 

8 
10 
12 
14 


3.59 
4.05 
4.12 
4.18 
4.25 


4.14 
4.20 
4.27 
4.34 

4.40 


4.20 
4.86 
4.43 
4.50 
4.57 


4.45 
4.52 
4.59 
5.06 


5.01 
5.08 
5 15 


5.17 
5. 24 


5.33 


5.50 
5.57 
6.05 
6.13 
6.21 


6.06 
6.14 
6.21 
6.29 
6.37 


6.28 
6.30 
6.38 
6.46 
6.54 


5.40 
5.48 
5.56 
6.04 


5 82 
5.39 
5.47 


5.22 
5,30 


5.18 


16 


4.31 


4.47 


5.04 


5.21 


5.38 


5.55 


6.12 


6.29 


6.45 


7.02 


18 


4.38 


4.55 


5.11 


5.28 


5 46 


6.03 


6.20 


6.37 


6.54 


7.11 


20 


4.45 


5.02 


5.19 


5.36 


5.54 


6.11 


6.29 


6.46 


7,03 


7-19 


22 


4.53 


5.10 


5.27 


5.45 


6.02 


6 20 


6.37 


6.55 


7.12 


7,29 


—24 


5.00 


5.18 


5. 36 


5.53 


6,11 


6.29 


6.47 


7.04 


7.21 


7.38 




117° 


iir 


111° 


108° 


105° 


102° 


99° 


96° 


93° 


90° 



139 



Azimuth and Hour Angle, for Latitude 




and Declination —Table Vl. 


LATITUDE 48^. 


Declination. 


^ziiynTJTHis. 


15° 


20° 


25° 


30° 


35° 40° 


45° 


50° 


55° 


60° 




H.M. 


H. M. 


H. M. 


H. M. 


H.M. H.M. 


H. M. 


H. M. 


H.M. 


H. M. 


+24° 


.27 


.37 


.46 


.57 


1.07 1.19 


1.31 


1.44 


1.59 


2.15 


22 


.29 


.39 


.49 


1.00 


1.11 , 1.24 


1.35 


1.50 


2.05 


2.22 


.] 20 


0.30 


0.41 


0.52 


1.03 


1.15 


1.28 


1.41 


1 56 


2.12 


2.29 


-S 18 


.32 


.43 


.54 


1.06 


1.19 


1.32 


1 46 


2.01 


2.18 


2.35 


B 16 


.33 


.45 


.57 


l.]0 


1.23 


1 37 


1.51 


2.07 


2.23 


2.42 


■? 14 


.35 


.47 


1.00 


1.13 


1.26 


1.41 


1.56 


2.12 


2 29 


2.48 


2 12 


.37 


.49 


1.02 


1.16 


1.30 


1.45 


2.00 


2.17 


2.35 


2.54 


ii 10 


.38 


0.51 


1.05 


1.19 


1.33 


1.48 


2.05 


2.22 


2 40 


3.00 


•1 8 


.39 


.53 


1.07 


1.21 


1.37 


1.52 


2.09 


2 27 


2.46 


3.06 


^ 6 


.41 


.55 


1.09 


1.24 


1.40 


1.56 


2.U 


2 32 


2.. 51 


3.11 


4 


.42 


.57 


].12 


1.27 


1.43 


2 00 


2.18 


2.37 


2.56 


3.17 


+ 2 


.44 


.59 


1.14 


1.30 


1.47 


2.04 


2.22 


2.41 


3.02 


3.23 





0.45 


1.01 


1.16 


1.33 


1.50 2.08 


2.26 


2 46 


3.07 


3.29 


— 2 


.46 


1.02 


1.19 


1.36 


1.53 


2.12 


2 31 


2. .51 


3.12 


3.34 


4 


.48 


1.04 


121 


1.39 


1.57 


2.15 


2.35 


2.56 


3.17 


3.40 


o 6 


.49 


1.06 


1.24 


1.41 


2.00 


2 19 


2.39 


3.01 


3 23 


3.46 


^ 8 


.51 


1.08 


1.26 


1.44 


2.03 


2.23 


2.44 


3.05 


3.28 


3 52 


•2 10 


0.52 


1.10 


1.28 


1.47 


2.07 


2 27 


2.48 


3 10 


3.33 


3.57 


i u 


.53 


1.12 


1.31 


1.50 


2.10 


2.31 


2.53 


3 15 


3.39 


4.03 


.55 


1.14 


1.33 


1.53 


2.14 


2.35 


2 57 


3.20 


3.44 


4.09 


^ 16 


.57 


1.16 


1.36 


1..56 


2.17 


2.39 


3.02 


3.26 


3.50 


4.16 


^ 18 
'P 20 


.58 
1.00 


1.18 
1.20 


1.38 
1.41 


1.59 
2.03 


2.21 
2.25 


2.43 

2.48 


3.07 
3 12 


3.31 
3.36 


3.56 
4. 02 


4.22 


4 29 


22 


1.01 


1.22 


1.44 


2.06 


2 29 


2.52 


3.18 


3.42 


4.0^ 


4 35 


—24 


1.03 


1.25 


1.47 


2.09 


2.33 


2.56 


3.22 


3.48 


4.10 


4.42 


















130° 


125° 


120° 



140 



Table VI.— Continued. 


LATITUDE 48^. 






.A-Zinv^TJTHIS- 


Declination. 












1 














63^ 


66° 


69° 


72° : 75° 


78° 


81° 


84° 


87° 


90° 




H. M. 


H. M. 


H. M. 


H. M. ; H. M. 

1 


H. M. 


H. M. 


H. M. 


H . .M . 


H. .M. 


+24° 


2.25 


2.36 


2.47 


2.59 3.12 


3.25 


3.40 


3.54 


4.10 


4.25 


22 


2.32 


2.44 


2.55 


3.08 3.20 


3.34 


3.48 


4.03 


4.19 


4.35 


20 


2.40 


2.51 


3.03 ! 3.16 3.28 


3.43 


3.57 


4.12 


4.28 


4.43 


18 


2.46 


2.58 


3.10 


3.23 3.36 


3.51 


4.05 


4.20 


4.36 


4.52 


16 


2.53 ; 3.05 


3.18 


3.31 3.44 


3.59 


4.13 


4.29 


4.44 


5.00 


14 


3.00 


3.12 


3.25 


3.38 3.51 


4.06 


4.21 


4.36 


4.52 


5.08 


12 


3.03 


3 18 


3.32 


3.45 j 3.59 


4.14 


4.29 


4.44 


5 00 


5.16 


10 


3.12 


3.25 


3.38 


3.52 : 4.06 


4.21 


4.36 


4.52 


5.07 


5.23 


8 


3.18 


3.31 


3.45 


3.59 ; 4.13 


4.28 


4.43 


4. 59 


5.15 


5.31 


6 


3.24 


3.38 


3.51 


4.06 1 4.20 


4.35 


4.51 


5.06 


5 22 


5.38 


4 


3.30 


3.44 


3.58 


4.12 


4.27 


4.42 


4.58 


5.14 


5.29 


5.46 


+ 2 


3.. 36 


3.50 


4.04 


4.13 


4.33 


4.49 


5.05 


5.21 


5.37 


5.53 





3.42 


3.56 


4.11 


4.26 


4.40 


4.56 


5.12 


5.28 


5.44 


6.00 




— 2 
4 


3.48 
3.54 


4.02 
4.09 


4 17 
4. 24 


4.32 
4.39 


4.47 
4.54 


5.03 
5.10 


5.19 
5.26 


5.35 
5.42 


5.51 


6.07 
6.14 


5.58 


6 
8 
10 
12 
14 
16 


4.00 
4.06 
4.12 
4.19 
4.25 
4.31 


4.15 
4.2] 
4.28 
4.34 
4.41 


4.30 
4.37 
4.43 
4.50 


4.45 
4.52 
4.59 


5.01 

5.08 
5.15 


5.17 
5.24 


5.33 


5.49 
5.57 
6.04 
6.11 
6.19 
6.27 


6.06 
6.13 
6.20 
6.28 
6.36 
6.44 


6.22 
6.29 
6.37 
6.44 
6.52 
7.00 


5.40 
5.48 
5.55 
6.03 
6.10 


5.31 
5.39 
5.46 
5.54 


5.06 
5.13 
5.20 


5.22 
5.29 
5.36 


4.57 
5.04 


4.47 
























18 


4.38 


4.54 


5.11 


5.28 


5.44 


6.01 


6.18 


6.35 


6.52 


7.08 


20 


4.45 


5.02 


5.18 


5.35 


5.52 


6.10 


6.27 


6.44 


7.00 


7 17 


22 


4.. 52 


5.09 


5.26 


5.43 


6.00 


6 18 


6.35 


6.52 


7.09 


7.25 


—24 


4.59 


5.17 


5.34 


5.52 


6.09 


6.27 


6.44 


7.01 


7.18 


7.35 




117° 


114° 


111° 


108° 


105° 


102° 


99° 


96° 


93° 


90° 



141 



TABLE VII. 



A table of differences of local and standard time, on the 
great lakes, for reducing standard tinie to the mean local time 
of the places mentioned. The sign prefixed, indicates the 
manner of applying the correction, to the standard time, 
the + sign meaning addition to, and the — sign meaning 
subtraction from standard time to get local mean time. 



LAKE ONTARIO. 

75=^ Long. 

EASTERN STANDARD. 

M. S. 

Sackett's Harbor — 4.36 

Stony Point — 5.12 

Fair Haven. — 6.52 

Big Sodus — 7.56 

Genesee —10.24 

Oak Orchard —12 .48 

30 Mile Point —13.56 

Fort Niagara —16.16 

Kingston.. — 5.54 

Desoronto , — 8.08 

BellviUe — 9.32 

Weller's Bay —10.44 

Coburg —12.56 

Port Hope —13 . 20 

Frenchman's Bay — 16 . 08 

Toronto —17.56 

Fort Dalhousie —17.24 



LAKE ERIE. 

75° Long. 

EASTERN STANDARD. 

M. S. 

Buffalo —15 .36 

Dunkirk —17.24 

Erie —20.16 

Ashtabula —23.12 

Fairport —25.08 

Cleveland —26.48 

Black River —28.44 

Cedar Point —30.48 

Sandusky Bay —30.52 

Green Island —31.28 

Turtle Island —33.32 

Detroit River (mouth) —33 . 32 

Detroit City —27.52 

Fort Colbourne — 17.16 

Fort Maitland —18.40 

Long Point (east end) —20 36 

Port Burwell —23.36 

Fort Stanley. —24.52 

Pelee Spit —30.32 

Middle Island —30.40 

Kingsville —30.56 

Amherst Bay — 32.52 



142 



TABLE VII.- 


-Continued. 




LAKE HURON. 

90° Long. 

CENTRAL STANDARD. 

M. S. 

BeUe Isle (Saitit Clair) +28. 12 

Fort Gratiot +30.20 


Grand Traverse 


.. +17.52 


South Manitou 


.. +15.40 


Manistee 

White River 


.. +14.36 
. +14.20 


Aluskegon 

Grand Haven. 


.. +14.40 
.. +15.00 


Fort Sanilac +29 .52 

Sand Beach +29.40 


St. Joseph 

Michigan City 

Chicago 


.. +14.04 
.. +12.24 
.. + 9.32 


Point of Barques +29.00 


Charity Island +26 . 16 

Saginaw River (mouth) +24.36 

Sturgeon Point +26.56 

Thunder Bay Island +27 .24 

Presque Isle +26.08 


Kenosha 


.. + 8.44 


Racine.. 


.. +8.52 


Milwaukee 

Sheboygan. 


.. + 8.32 
+ 9.12 


Sturgeon Bay Canal 

Pilot Island 


.. + 9.44 
.. +12.20 


Spectacle Reef +23.28 


Cheboygan +22.20 

Detour +24.24 


Escanaba 


+ 11 52 


Chamber's Island. 


. +10 32 


CANADIAN SIDE. 

75° Long. 

M. S. 

Goderich —26.12 


Menominee 


.. + 9.40 




OR. 

mD. 

M. S. 

.. +22.32 

.. +20.12 


LAKE SUPER 

00^ Long. 
CENTRAL stand; 

St. Mary's Falls 

White Fish Point 


South Hampton — 25.32 

Michael's Point .'. —27 . 44 

Great Duck Island —31.44 

Owen Sound —23.44 


CoUingwood — 20 . 08 


Grand Island 


.. +13.16 


Whiskey Island —19 . 32 

FrenchRiver —23.40 

Manitowaning — 27.12 

Thessdon River —23.40 


Marquette 


.. +10.28 


Stanard's Rock 

Copper Harbor 

Ontonagon 


.. +11.08 

.. + 8.32 

. +2.40 




Devil's Island 

St. Louis River ... . 


.. —3.12 

.. — 8.04 


LAKE MICHIGAN. 

90® Long. 

CENTRAL STANDARD. 

M. S. 

McGulpin's Point..,. +20.56 

Skilligallee +19.20 


Duluth 


... — 8.20 


Grand Marias 


.. — 1.20 


Isle Royal 

CANADA. 
Agate Island 


.. 4- 5.00 

M. S. 

.. +15.52 


Lamb Island 


.. + 6.32 


Beaver Island + 17 . 44 

Little Traverse +20 08 


Thunder Cape 


.. + 3.20 


Port Arthur 


.. + 3.08 


South Fox Island +16.40 


Victoria Island 


.. + 2.36 



143 



Table 


Of 


Chords to 


Radius Unity, for 1 








Protracting- 


-VII 


■ 













1° 


2° 


3° 


4° 


5° 


6° 


7° 


8° 


9° 




0' 


.jooo 


.0175 


.0349 


.0.524 


.0698 


.0872 


.0147 


.0 


.1395 


.1.569 


0' 


10 


.0027 


.0204 


.0378 


.0553 


.0727 


.0901 


.1076 


.1250 


.1424 


.1598 


10 


20 


.0058 


.0233 


.0407 


.0582 


.0756 


.0931 


.1105 


.1279 


.14.53 


.1627 


20 


30 


.0087 


.0262 


.0436 


.0611 


.0785 


.0960 


.1134 


.1308 


.1482 


.16.56 


30 


40 


.0116 


.0291 


.0465 


.0640 


.0814 


.0989 


.1163 


.1337 


.1511 


.1685 


40 


50 


.0145 


.0320 


.0494 


.0669 


.0843 


.1018 


.1192 


.1366 


.1540 


.1714 


50 




10° 


11° 


12° 


13° 


14° 


15° 


16° 


17° 


18° 


1S° 




0' 


.1743 


.1917 


.2091 


.2264 


.2437 


.2611 


.2783 


.29.56 


.3129 


.3301 


0' 


10 


.1772 


.1946 


.2119 


.2293 


.2466 


.2639 


.2812 


.2985 


.3157 


. 3330 


10 


20 


.1801 


.1975 


.2148 


.2322 


.2495 


.2668 


.2841 


.3014 


.3186 


.3358 


20 


30 


.1830 


.2004 


.2177 


.2351 


.2524 


.2697 


.2870 


.3042 


.3215 


.3387 


30 


40 


.1859 


.2033 


.2206 


.2380 


.2.5.53 


.2726 


.2899 


.3071 


.3244 


.3416 


40 


60 


.1888 


.2062 


.2235 


.2409 


.2582 


.2755 


.2927 


.3100 


.3272 


.3444 


50 




20° 


21° 


22° 


23° 


24° 


25° 


26° 


27° 


28° 


29° 




0' 


.3473 


.3645 


.3816 


.3987 


.4158 


.4329 


.4499 


.4669 


.4838 


..5008 


0' 


10 


.3502 


.3673 


.3845 


.4016 


.4187 


.4357 


.4527 


.4697 


.4867 


..5036 


10 


20 


.3.530 


.3702 


.3873 


.4044 


.4215 


.4386 


.5556 


.4725 


.4895 


.5064 


20 


30 


.3559 


.3730 


.3902 


.4073 


.4244 


.4414 


.4584 


.4754 


.4923 


.5092 


30 


40 


.3587 


.3759 


.3930 


.4101 


.4272 


.4442 


.4612 


.4782 


.4951 


.5120 


40 


50 


.3616 


.3788 


.3959 


.4130 


.4300 


.4471 


.4641 


.4810 


.4979 


.5148 


50 




30° 


31° 


32° 


33° 


34° 


35° 


36° 


37° 


38° 


39° 




0' 


.5176 


.5345 


.5513 


.5680 


.5847 


.6014 


.6180 


.6346 


.6511 


.6676 


0' 


10 


.5204 


.5373 


.5541 


.5708 


.5875 


.6042 


.6208 


.6374 


.6539 


.6704 


10 


20 


.5233 


.5401 


.5569 


.5736 


..5903 


.6070 


.6236 


.6401 


.6566 


6731 


20 


30 


.5261 


.5429 


.5597 


.5764 


.5931 


.6097 


.6263 


.6429 


.6594 


.6778 


30 


40 


..5289 


.5457 


..5625 


.5792 


.5959 


.6125 


.6291 


.6456 


.6621 


.6786 


40 


50 


.5317 


.5485 


.5652 


.5820 


.5986 


.6153 


.6318 


.6484 


.6649 


.6813 


50 




40° 


41° 


42° 


43° 


44° 


45° 


46° 


47° 


48° 


49° 




0' 


.6840 


.7004 


.7167 


.7330 


.7492 


.7654 


.7815 


.7975 


.8135 


.8294 


0' 


10 


.6868 


.7031 


. 7195 


. 7357 


.7519 


.7681 


.7841 


.8002 


.8161 


.8320 


10 


20 


.6895 


.7059 


.7222 


.7384 


.7.546 


.7707 


.7868 


.8028 


.8188 


.8347 


20 


30 


.6922 


.7086 


.7249 


.74J1 


.7573 


.7734 


.7895 


.8055 


.8214 


.8373 


30 


40 


.6950 


.7113 


.7276 


.7438 


7600 


7761 


.7922 


.8082 


.8241 


.840') 


40 


50 


.6977 


.7140 


.7303 


.7465 .7627 


.7788 


.7948 


.8108 


.8267 


.8426 


50 



144 



Table VIII.— Continued. 




50° 


51° 


52° 


53° 


54° 


55° 


56° 


57° 


58° 


59° 




0' 


.8452 


.8610 


.8767 


.8924 


.9080 


.9235 


.9389 


.9543 


9606 


.9848 


0' 


10 


.8479 


.8606 


.8794 


.89r.O 


.9106 


.9261 


.9415 


.9569 


.9722 


.9874 


10 


20 


.•8505 


.8663 


.8820 


.8976 


.9132 


.9287 


.9441 


.9594 


.9747 


.9899 


20 


30 


.8531 


.'8689 


.8846 


.9002 


.9157 


.9312 


.9466 


.9620 


.9772 


.9924 


30 


40 


.8558 


.8715 


.8872 


.9028 


.9183 


.9338 


.9192 


.9645 


.<-798 


.9950 


40 


50 


.8584 


.8741 


.8898 


.9054 


.9209 


.9364 


.9518 


.9671 


.9823 


.9975 


50 




60° 


61° 


62° 


63° 


64° 


65° 


66° 


67° 


68° 


6S° 




0' 


1.0000 


1.0157 


1.0,301 


1.0450 


1.0598 


1.0746 


1.0893 


1.1C39 


1.1184 


1.1328 


0' 


10 


1.0025 


1.0176 


1.032611.04751.06231.0771 


1.09171.1063 


1 . 1208 


i.l352 


10 


20 


1.0050 


1.0201 


1.03.51! 1.0500 1.06481.0795 


1 094211.1087 


1 1232 


1.1376 


20 


30 


1 0075 


1.0226 


1 .0375 '1 .0.524 1 .0672 1 .0820 


1.09661.1111 


1.1256 


1.1400 


30 


40 


i.oini 


1.0251 


1.0400 1.0549 1.06971 0844 


1.09901.1136 


1.1280 


1.1424 


40 


60 


1.0126 


1.0.!76 


1.0425 


1.05741.07211.0868 


1.1014 


1.1160 


1.1304 


L.1448 


50 




70° 


71° 


72° 


73° 74° 


75° 


76° 


77° 


78^ 


79° 




0' 


1.1472 


1.1614 


1.1756 


l.i896'l. 20361. 2175 


1.2313 


1.2450 


1.2.586 


1.2722 


0' 


10 


1.1495 


1,1638 


1.1779 


1.19201.20601.2198 


1.23361 2473 


1.2609 


1.2744 


10 


20 


1.1519 


1.1661 


1.1803 


1.194311.20831.2221 


1.23591 2496 


1 2632 


1.2766 


20 


30 


1.1543 


1.1685 


1.1826 


1.1966;i.2106 


1.2244 


1.2382 


1.2518 


1.2654 


1.2789 


30 


40 


1.1.567 


1.1709 


1.1850 


1.19901.2129 


1.2267 


1.2405 


1.2541 


1.2677 


1.2811 


40 


50 


1.1590 


1-1732 


1.1873 


1.20131.2152 


1.2290 


1.2428 


1.2564 


1.2699 


1.2833 


50 




80° 


81° 


82° 


83° 


84° 


85° 


86° 


87° 


88° 


89° 




0' 


1.2856 


1.2989 


1.3121 


1.3252 


1.3383 


1.3512 


1 3640 


1.3767 


1.3893 


1.4018 


0' 


10 


1.2878 


1.3011 


1.3143 


1. 3274 :x. 3404 


1.3533 


1 3661 


1.3788 


1.3914 


1.4039 


10 


20 


1.2900 


1.3033 


1.3165 


1. 3296 !x. 3426 


1.3555 


1.3682 


1.3809 


1.3935 


1.4060 


20 


30 


1.2922 


1.3055 


1.3187 


1. 3318 il. 3447 


1.3.576 


1.3704 


1 3830 


1.3956 


1.4080 


30 


40 


1.2945 


1.3077 


1.3209 


1 33391.3469 


1.3597 


1.3725 


1.3851 


1.3977 


1.4101 


40 


50 


1.2967 


1.3099 


1.3231 


1.33611.3490 


1.3619 


1.3746 


1.3872 


1.3997 


1.4122 


50 




90° 


91° 


92° 


9o° 

1.4508 


94° 


95° 


96° 


97° 


98° 


990 




0' 


1.4142 


• 
1.4265 


1.4386 


1.4627 


1.4745^1.4863 


1.4979,1.5094 


1.5208 


0' 


10 


1.4162 


1 .4285 1. 4406 1 .4528 1 .464.7 1 .4765 1 .4883 1 .4998 


1.51131.5227 


10 


20 


1.41831.43051.44261.45481.46671.47851.49021.5017 


1.51321.5246 


20 


30 


1 .4203 1 .4325 1 .4446 1 .4568 1 .46861 .4804 1 .4921 1 .5037 


1.. 51.51 1.5265 


30 


40 


1 .4221 1 .4346 1 .4467 1 .4588 1 .4706 1 .4824 1 .4940 1 .5056 


1.51701.5283 


40 


50 


1 .4245 1 .4366 1 .4487 1 .4608 1 .4726 1 .4843 1 .4960 1 .5074 1 .5189 1 .5302 

! i ! ' 1 1 1 1 1 


50 



145 











TABLE 


IX 


■ 










MERIDIANAL PARTS. 






M. 


0° 


P 


2° 


2P 


40 


5° 


Qo 


70 


8° 


9° 


10° 


IP 


12° 


M. 








60 


120 


180 


240 


300 


361 


421 


482 


542 


603 


664 


725 





2 


2 


62 


122 


182 


242 


302 


363 


423 


4S4 


544 


(•(5 


666 


727 


2 


4 


4 


64 


124 


184 


244 


304 


365 


425 


486 


546 


607 


66^ 


729 


4 


6 


6 


66 


126 


1% 


246 


306 


367 


427 


488 


548 


609 


670 


731 


6 


8 


8 


68 


128 


188 


248 


308 


369 


4^9 


490 


550 


611 


672 


734 


8 


10 


10 


70 


130 


190 


250 


310 


371 


431 


492 


552 


613 


674 


736 


10 


12 12 


72 


132 


192 


252 


312 


373 


4M3 


4 4 


554 


615 


676 


7:^8 


12 


14 


14 


74 


134 


194 


254 


3] 4 


375 


435 


496 


556 


617 


678 


740 


14 


16 


16 


76 


136 


196 


256 


316 


377 


437 


498 


558 


619 


6S0 


742 


16 


18 


18 


78 


138 


198 


258 


318 


379 


439 


500 


560 


621 


682 


744 


18 


20 


20 


80 


140 


200 


260 


320 


381 


441 


502 


562 


623 


684 


746 


20 


22 


22 


82 


142 


202 


262 


322 


383 


443 


504 


565 


625 


687 


748 


22 


24 


24 


84 144 


204 


264 


324 


385 


445 


506 


567 


627 


689 


750 


24 


26 


26 


86 


146 


206 


266 


326 


387 


447 


508 


569 


629 


691 


752 


26 


28 


28 


88 


148 


208 


268 


328 


389 


449 


510 


571 


632 


693 


754 


28 


30 


30 


90 


150 


210 


270 


331 


391 


451 


512 


573 


634 


695 


756 


30 


32 


32 


92 


152 


212 


272 


333 


393 


4.^3 


514 


575 


636 


697 


758 


32 


34 


34 


94 


154 


214 


274 


335 


395 


455 


516 


577 


638 


699 


760 


34 


36 


36 


96 


156 


216 


276 


3a7 


397 


457 


518 


579 


640 


701 


762 


36 


38 


38 


98 


158 


218 


278 


339 


399 


459 


520 


581 


642 


703 


764 


38 


40 


40 


100 


160 


220 


280 


341 


401 


461 


522 


583 


644 


705 


766 


40 


42 


42 


102 


162 


222 


2*^2 


343 


403 


4r.3 


524 


685 


646 


707 


768 


42 


44 


44 


104 


164 


224 


284 


345 


405 


465 


526 


587 


648 


709 


770 


44 


46 


46 


106 


166 


226 


286 


347 


407 


467 


528 


589 


650 


711 


772 


46 


48 


48 


108 


168 


228 


288 


349 


409 


469 


530 


591 


652 


713 


774 


48 


50 


50 


no 


170 


230 


290 


351 


411 


471 


532 


593 


654 


715 


777 


50 


52 


52 


112 


172 


232 


292 


353 


413 


473 


534 


595 


656 


717 


779 


62 


54 


54 


114 


174 


234 


294 


355 


415 


476 


536 


597 


658 


719 


781 


64 


56 


56 


116 


176 


236 


296 


357 


417 


478 


538 


599 


660 


721 


783 


66 


68 


58 


118 


178 


238 


298 


359 


419 


480 


640 


601 


662 


723 


785 


68 


M. 


0^ 


1^ 


2° 


3° 


4^ 


5° 


6° 


70 


9P 


9° 


10° 


IP 


12° 


M. 



146 



TABLE IX.-Continued. 


M. 


13° 


14° 


15° 


16° 


17° 


18° 


19° 


20° 


21° 


22° 


M. 




2 
4 
6 

8 

10 
12 
14 

16 

18 

20 
22 
24 
26 

28 

30 
32 
34 

36 

38 

40 
42 
44 
46 
48 

50 
52 

54 
56 

58 


787 
789 
791 
793 
795 

797 
799 
801 
803 
805 

807 
809 
811 
813 
816 

818 
820 
822 
824 
826 

828 
830 
832 
834 
836 

838 
840 
842 
844 
846 


848 
851 
853 
855 
857 

859 
861 
863 
865 
867 

869 
871 
873 
875 
877 

879 
882 
884 
886 
888 

890 
892 
894 
896 
898 

900 
902 
904 
906 
908 


910 
913 
915 
917 
919 

921 
923 
925 
927 
929 

931 
933 
935 
937 
939 

942 
944 
946 
948 
950 

952 
954 
956 

958 
960 

962 
964 
966 
969 
971 


973 
975 

977 
979 
981 

983 
985 
987 
989 
991 

994 

996 

998 

1000 

1002 

1004 
1006 
1008 
1010 
1012 

1014 
1016 
1019 
1021 
1023 

1025 
1027 
1029 
1031 
1033 


1035 
1037 
1039 
1042 
1044 

1046 
1048 
1050 
1052 
1054 

1056 
1058 
1060 
1063 
1065 

1067 
1069 
1071 
1073 
1075 

1077 
1079 
1081 

1084 
1086 

1088 
1090 
1092 
1094 
1096 


1098 
1100 
1102 
1105 
1107 

1109 

nil 

1113 
1115 
1117 

1119 
1121 
1123 
1126 
1128 

1130 
1132 
1134 
1136 
1138 

1140 
1142 
1145 
1147 
1149 

1151 
1153 
1155 
1157 
1159 


1161 
1161 
1166 
1168 
1170 

1172 
1174 
1176 
1178 
1181 

1183 
1185 
1187 
1189 
1191 

1193 
1195 
1198 
1200 
1202 

1204 
1206 
1208 
1210 
1212 

1215 
1217 
1219 
1221 
1223 


1225 

1227 
1229 
1232 
1234 

1236 
1238 
1240 
1242 
1245 

1247 
1249 
1251 
1253 
1255 

1257 
1259 
1261 
1264 
1266 

1268 
1270 
1272 
1274 
1276 

1278 
1281 
1283 

1285 
1287 


1289 
1291 
1293 
1296 
1298 

1300 
1302 
1304 
1306 
130S 

1311 
1313 
1315 
1317 
1319 

1321 
1324 
1326 
1328 
1330 

1332 
1334 
1336 
1339 
1341 

1343 
1345 
1347 
1349 
1352 


1354 
1356 
1358 
1360 
1362 

1364 
1367 
1369 
1371 
1373 

1375 
1377 
1380 
1382 
1384 

1386 
1388 
1390 
1393 
1395 

1397 
1399 
1401 
1403 
1406 

1408 
1410 
1412 
1414 
1416 



2 
4 
6 

8 

10 
12 
14 
16 
18 

20 

22 
24 
26 

28 

30 
32 
34 
36 
38 

40 
42 
44 
46 
48 

50 
52 

54 
56 
58 


M. 


13° 


14° 


15° 


16° 


17° 


18° 


19° 


20° 


21° 


22° 


M. 



147 



TABLE IX.-Continued. 


M. 


23° 


21° 


25° 


26° 


27° 


28° 


29° 


30° 


31° 


32° 


M. 



2 
4 

6 
8 

10 
12 
14 
16 

18 

20 
22 
24 
26 
28 

30 
32 
34 

36 
38 

40 
42 
44 
46 
48 

50 
52 
54 
56 
58 


1419 
1421 
1423 
1425 
1427 

1430 
1432 
1434 
1436 
1438 

1440 
1443 
1445 
1447 
1449 

1451 
1453 
1456 
1458 
1460 

1462 
1464 
1467 
1469 
1471 

1473 
1475 
1477 
1480 
1482 


1484 
1486 
1488 
1491 
1493 

1495 
1497 
1499 
1502 
1504 

1506 
1508 
L510 
1513 
1515 

1517 
1519 
1521 
1524 
1526 

1528 
1530 
1532 
1535 
1537 

1539 
1541 
1543 
1546 
1548 


1550 
1552 
1554 
1557 
1559 

1561 
1563 
1565 
1568 
1570 

1572 
1574 
1577 
1579 
1581 

1583 
lo85 
1588 
1590 
1592 

1594 
1596 
1599 
1601 
1603 

1605 
1608 
1610 
1612 
1614 


1616 
1619 
1621 
1623 
1625 

1628 
1630 
1632 
1634 
1637 

1639 
1641 
1643 
1645 
1648 

1650 
1652 
1654 
1657 
1659 

1661 
1663 
1666 
1668 
1670 

1672 
1675 
1677 
1679 
1681 


1684 
16M6 
1688 
1690 
1693 

1695 
1697 
1699 
1701 
1704 

1706 
1708 
1711 
1713 
1715 

1717 
1720 
1722 
1724 
1726 

1729 
1731 
1733 
1735 
1738 

1740 
1742 
1744 
1747 
1749 


1751 
1753 
1756 
1758 
1760 

1762 
1765 
1767 
1769 
1772 

1774 
1776 

1778 
1781 
1783 

1785 
1787 
1790 
1792 
1794 

1797 
1799 
1801 
1803 
1806 

1808 
1810 
1813 
1815 
1817 


1819 
1822 
1824 
1826 
1829 

1831 
1833 
1835 
1838 
1840 

1842 

1845 
1847 
1849 
1852 

1854 
1856 
1858 
1861 
1863 

1865 
1868 
1870 
1872 
1875 

1877 
1879 
1881 
1884 
1886 


1888 
1891 
1893 
1895 
1898 

1900 
1902 
1905 
1907 
1909 

1912 
1914 
1916 
1918 
1921 

1923 
1925 

1928 
1930 
1932 

1935 
1937 
1939 
1942 
1944 

1946 
1949 
1951 
1953 
1956 


1958 
1960 
1963 
1965 
1967 

1970 
1972 
1974 
1977 
1979 

19^1 
1984 
1986 
1988 
1991 

1993 
1995 
1998 
2000 
2C02 

2005 
2007 
2010 
2012 
2014 

2017 
2019 
2021 
2024 
2026 


2028 
2031 
2033 
2035 
2038 

2040 
2043 
2045 
2047 
2050 

2052 
2054 
2057 
2059 
2061 

2064 
2066 
2069 
2071 
2073 

2076 
2078 
2080 
2083 
2085 

2088 
2090 
2092 
2095 
2097 



2 

4 
6 

8 

10 
12 
14 
16 
18 

20 
22 
24 
26 
28 

30 
32 
34 
36 
38 

40 
42 
44 
46 
48 

50 
52 

54 
56 
58 


M. 


23° 


24° 


25° 


26° 


27° 23° 


29° 


30° 


31° 


32° 


M. 



148 









TABLE 


IX- 


-Continued. 








M. 


33" 


34° 


35° 


36° 


37° 


38° 


39° 


40° 


41° 


42° 


M. 





2100 


2171 


2244 


2318 


2393 


2468 


2545 


2623 


2702 


2782 





2 


2102 


2174 


2247 


2320 


2895 


2471 


2548 


2625 


2704 


2784 


2 


4 


2104 


2176 


2249 


2323 


2398 


2473 


2550 


2628 


2707 


2787 


4 


6 


2107 


2179 


2252 


2325 


2400 


2476 


2553 


2631 


2710 


2790 


6 


8 


2109 


2181 


2254 


2328 


2403 


2478 


2555 


2633 


2712 


2792 


8 


10 


2111 


2184 


2257 


2330 


2405 


2481 


2558 


2636 


2715 


2795 


10 


12 


2114 


2186 


2259 


2333 


2408 


2484 


2560 


2638 


2718 


2798 


12 


14 


2116 


2188 


2261 


2335 


2410 


2486 


2563 


2641 


2720 


2h01 


14 


16 


2119 


2191 


2264 


2338 


2413 


2489 


2566 


2644 


2723 


2803 


16 


18 


2121 


2193 


2266 


2340 


2415 


2491 


2568 


2646 


2726 


2806 


18 


20 


2123 


2196 


2269 


2343 


2418 


2494 


2571 


2649 


2728 


2809 


20 


22 


2126 


2198 


2271 


2345 


2420 


2496 


2573 


2651 


2731 


2811 


22 


24 


2128 


2200 


2274 


2348 


2423 


2499 


2576 


2654 


2-33 


2814 


24 


26 


2131 


2203 


2276 


2350 


2425 


2501 


2578 


2657 


2736 


2817 


26 


28 


2133 


2205 


2279 


2353 


2428 


2504 


2581 


2659 


2739 


2820 


28 


30 


2135 


2208 


2281 


2355 


2430 


2506 


2584 


2662 


2742 


2822 


30 


32 


2138 


2210 


2283 


2358 


2433 


2509 


2586 


2665 


2744 


2825 


32 


34 


2140 


2213 


2286 


2360 


2435 


2512 


25S9 


2667 


2747 


2828 


34 


36 


2143 


2215 


2288 


2363 


2438 


2514 


2591 


2670 


2750 


2830 


36 


38 


2145 


2217 


2291 


2365 


2440 


2517 


2594 


2673 


2752 


2833 


38 


40 


2147 


2220 


2293 


2368 


2443 


2519 


2597 


2675 


2755 


2«36 


40 


42 


2150 


2222 


2296 


2370 


2445 


2522 


2599 


2678 


2758 


2839 


42 


44 


2152 


2225 


2298 


2373 


2448 


2524 


2602 


2680 


2760 


2841 


44 


46 


2155 


2227 


2301 


2375 


2451 


2527 


2604 


2683 


2763 


2844 


46 


48 


2157 


2230 


2303 


2378 


2453 


2530 


2607 


2686 


2766 


2847 


48 


50 


2159 


2232 


2306 


2380 


2456 


2532 


2610 


2688 


2768 


2849 


50 


52 


2162 


2235 


2308 


2383 


2458 


2535 


2612 


2691 


2771 


2652 


52 


54 


2164 


2237 


2311 


2385 


2461 


2537 


2615 


2694 


2774 


2855 


54 


56 


2167 


2239 


2313 


2388 


2463 


2540 


2617 


2i'96 


2776 


2858 


56 


58 


2169 


2242 


2316 


2390 


2466 


2542 


2620 


2699 


2779 


2860 


58 


M. 


33° 


34° 


35° 


36° 


37° 


38° 


39° 


40° 


41° 


42° 


M. 



149 



TABLE iX.-Continued. 


M. 


43° 


44° 


45° 


46° 


47° 


48° 


49° 


50° 


51° 


52° 


M. 



2 
4 
6 

8 

10 
12 
14 
16 

18 

20 
22 
24 

26 

28 

30 
32 
34 
36 
38 

40 
42 

44 

46 
48 

50 
52 
54 
66 
58 


2863 
2866 
2869 

2871 
2874 

2877 
2880 
2882 
2885 
2888 

2891 
2893 
2896 
2899 
2902 

2904 
2907 
29 
2Q13 
2915 

2918 
2921 

2^24 
2926 
2929 

29^2 
2"35 
2937 
2^^0 
2943 


2946 
2949 
2951 
2954 
2957 

2960 
2963 
2965 
2968 
2971 

2974 
2976 
2979 
2982 
2985 

2988 
2991 
2993 
2996 
2999 

3002 
3005 
3007 
3010 
3013 

3016 
3019 
3021 

3024 
3027 


3030 
3033 
3036 
3038 
3041 

3044 
3047 
3050 
3053 
3055 

305'8 
3061 
3064 
3067 
3070 

3073 
3075 
30''8 
3081 
3084 

3087 
3090 
3093 
3095 
30i.8 

3101 
3104 
3107 
3110 
3113 


3116 
3118 
3i21 
3124 
3127 

3130 
3133 
3136 
3139 
3142 

3144 
3147 
3150 
3153 
3156 

3159 
3162 
3165 
3168 
3171 

3173 
3176 
3179 

3182 
3185 

3188 
3191 
3194 
3197 
3200 


3203 
3206 
3209 
3212 
3214 

3217 
3220 
3223 
3226 
3229 

3232 
3235 
3238 
3241 
3244 

3247 

3250 
3253 
3256 
3259 

3262 
3265 
3268 
3271 
3274 

3277 
3280 
3283 

3286 
3239 


3292 
3295 
3298 
3301 
3303 

3306 
3309 
3312 
3316 
3319 

3322 
3325 
3328 
3331 
3334 

3337 
3340 
3343 
3346 
3349 

3352 
3355 
3358 
3361 
3364 

3367 
3370 
3373 
3376 
3379 


3382 
3385 
3388 
3391 
3394 

3397 
3400 
3403 
3407 
3410 

3413 
3416 
3419 
3422 
3425 

3428 
3431 
3434 
3*37 
3440 

3443 
3447 
3450 
3453 
3*56 

3459 
3*62 
3*65 
3*68 
3471 


3474 
3478 
3481 
3484 
3487 

3490 
3493 
3496 
3499 
3503 

3506 
3509 
3512 
3515 
3518 

3521 
3525 
3528 
3531 
3534 

3537 
3540 
3543 
3547 
3550 

3553 
3556 
3559 
3^62 
3566 


3569 
3572 
3575 
3578 
3582 

3585 
3588 
3591 
3594 
3598 

3601 
3604 
3607 
3610 
3614 

3617 
3620 
3623 
3626 
3630 

3633 
3636 
3639 
3643 
3646 

3649 
3652 
3655 
3659 
3662 


3665 
3668 
3672 
3675 
3678 

3681 
3685 
3688 
3691 
3695 

3698 
3701 
3''04 
3708 
3711 

3714 
3717 
3721 
3724 
3727 

3731 
3734 
3737 
37*1 
3744 

3747 
3750 
3754 
3757 
3760 



2 
4 
6 
8 

10 
12 
14 
16 
18 

20 
22 
24 
26 

28 

30 
32 
34 
36 
38 

40 
42 
44 
46 
48 

50 
52 

54 
56 
58 


M. 


43° 


44° 


45° 


46° 


47° 


48° 


49° 


50° 


51° 


52° 


M. 



150 









TABLE 


IX- 


-Continued. 






M. 


53" 


54° 


55° 


56° 


57° 


58° 


59° 


60° 


61° 


62° 


M. 





3764 


3865 


3968 


4074 


4183 


4294 


4409 


4527 


4649 


4775 





2 


3767 


3«68 


3971 


4077 


4186 


4298 


4413 


4531 


4653 


4779 


2 


4 


3770 


3«71 


3975 


4081 


4190 


4302 


4417 


4535 


4657 


4784 


4 


6 


3774 


3875 


3978 


4085 


4194 


4306 


4421 


4539 


4662 


4788 


6 


8 


3777 


3878 


3982 


4088 


4197 


4309 


4425 


4543 


4666 


4792 


8 


10 


3780 


3882 


3985 


4092 


4201 


4313 


4429 


4547 


4670 


4796 


10 


12 


3784 


3b85 


3989 


4095 


4205 


4317 


4433 


4551 


4674 


4801 


12 


14 


3787 


3889 


3992 


4f99 


4208 


4321 


4436 


4555 


4678 


4805 


14 


16 


3790 


3892 


3996 


4103 


4212 


4325 


4440 


4559 


4682 


4809 


16 


18 


3794 


3895 


3999 


4106 


4216 


4328 


4444 


4564 


4687 


4814 


18 


20 


3797 


3899 


4003 


4110 


4220 


4332 


4448 


4568 


4691 


4818 


20 


22 


3800 


3902 


4006 


4113 


4223 


4336 


4452 


4572 


4695 


4822 


22 


24 


3804 


3906 


4010 


4117 


4227 


4340 


4456 


4576 


4699 


4826 


24 


26 


3807 


3909 


4014 


4121 


4231 


4344 


4460 


4580 


4703 


4831 


26 


28 


3811 


3913 


4017 


4124 


4234 


4347 


4464 


4584 


4707 


4835 


28 


30 


3814 


3916 


4021 


4128 


4238 


4351 


4468 


4588 


4712 


4839 


30 


32 


3817 


3919 


4024 


4132 


4242 


4355 


4472 


4592 


4716 


4844 


32 


34 


3821 


3923 


4028 


4135 


4246 


4359 


4476 


4596 


4720 


4848 


34 


36 


3824 


3926 


4031 


4139 


4249 


4363 


4480 


4600 


4724 


4852 


36 


38 


3827 


3930 


4035 


4142 


4253 


4367 


4484 


4604 


4728 


4857 


38 


40 


3«31 


3983 


4038 


4146 


4257 


4370 


4488 


4608 


4733 


4861 


40 


42 


3834 


3937 


4042 


4150 


4260 


4374 


4492 


4612 


4737 


4865 


42 


44 


3838 


3940 


4045 


4153 


4>64 


4378 


4499 


4616 


4741 


4870 


44 


46 


3841 


3944 


4049 


4157 


4268 


4382 


4495 


4620 


4745 


4874 


46 


48 


3844 


3947 


4052 


4161 


4272 


4386 


4503 


4625 


4750 


4879 


48 


50 


3848 


3951 


4056 


4164 


4275 


4390 


4507 


4629 


4754 


4883 


50 


52 


3851 


3954 


4060 


4168 


4279 


4894 


4511 


4633 


4758 


4887 


52 


54 


3854 


3958 


4063 


4172 


4283 


4398 


4515 


4637 


4762 


4892 


54 


56 


3858 


3961 


4067 


4175 


4287 


4401 


4519 


4641 


4766 


4896 


56 


58 


3861 


3964 


4070 


4179 


4291 


4405 


4523 


4645 


4771 


4901 


58 


M. 


53° 


54° 


55° 


56° 


57° 


58° 


59° 


60° 


61° 


62° 


M. 



151 









TABLE 


X- 


-Continued. 








M. 


63° 


64° 


65° 


66° 


67° 


68° 


69° 


70° 


71° 


72° 


M. 





4905 


5039 


5179 


5324 


5474 


5631 


5795 


5966 


6146 


6335 





2 


4909 


5044 


5184 


5328 


5479 


5636 


5800 


5972 


6152 


63^1 


2 


4 


4914 


5049 


5188 


5333 


5484 


5642 


5806 


5978 


6158 


6348 


4 


6 


4918 


5053 


5193 


5338 


5489 


5647 


5811 


5984 


6164 


6.'^-4 


6 


8 


4923 


5058 


5198 


5343 


5495 


5652 


5817 


5989 


6170 


6361 


8 


10 


4927 


5062 


5203 


5348 


5500 


5658 


5823 


5995 


6177 


6367 


10 


12 


4931 


5067 


5207 


5353 


5505 


5663 


5828 


6001 


6183 


6374 


12 


14 


4936 


5071 


5212 


5358 


5510 


5668 


5834 


6007 


6189 


6380 


14 


16 


4940 


5076 


5217 


5363 


5515 


5674 


5839 


6013 


6195 


63h7 


16 


18 


4945 


5081 


5222 


5368 


5520 


5679 


5845 


6019 


6201 


6394 


18 


20 


4^49 


5085 


5226 


5373 


5526 


5685 


5851 


6025 


6208 


6400 


20 


22 


4954 


5090 


5231 


5378 


5531 


5690 


5856 


6031 


6214 


6407 


22 


24 


4958 


5095 


5236 


5383 


5536 


5695 


5862 


6037 


6220 


6413 


24 


26 


4963 


5099 


5'241 


5388 


5541 


5701 


5868 


6043 


6226 


6420 


26 


28 


4967 


5104 


5246 


5393 


5546 


5706 


5874 


6049 


6233 


6427 


28 


30 


4972 


5108 


5250 


5398 


5552 


5712 


5879 


6055 


6239 


6433 


30 


32 


4976 


5113 


5>55 


5403 


5557 


5717 


5885 


6061 


6245 


6440 


32 


34 


4981 


5118 


5260 


5408 


5562 


5723 


5891 


6067 


6252 


6447 


34 


36 


4985 


5122 


5265 


5413 


5567 


5728 


5896 


6073 


6258 


6453 


36 


38 


4990 


5127 


5270 


5418 


5573 


5734 


5902 


6079 


6264 


6460 


38 


40 


4994 


5132 


5275 


5423 


5578 


5739 


5908 


6085 


6271 


6467 


40 


42 


4999 


5136 


5280 


5428 


5583 


5745 


5914 


6091 


6277 


6473 


42 


44 


5003 


5141 


5284 


5433 


5588 


5750 


5919 


6097 


6283 


6480 


44 


46 


5008 


5146 


5289 


5438 


5594 


5756 


5925 


6103 


6290 


6487 


46 


48 


5012 


5151 


5294 


5443 


5599 


5761 


5931 


6109 


6296 


6494 


48 


50 


5017 


5155 


5299 


5448 


5604 


5767 


5937 


6115 


6303 


6500 


50 


52 


5021 


5160 


5304 


5454 


5610 


5772 


5943 


6121 


6309 


6507 


52 


54 


5026 


5165 


5309 


5459 


5615 


577« 


5948 


6127 


6315 


6514 


64 


56 


5030 


5169 


5314 


5464 


5620 


5783 


5954 


6133 


6322 


6521 


66 


58 


5035 


5174 


5319 


5469 


5625 


5789 


5960 


6140 


6328 


6528 


58 


M. 


63° 


64° 


65° 


66° 


67° 


68° 


69° 


70° 


71° 


72° 


M. 



152 











TABLE 


X. 










Showing it 


e length of 1 


minute of longitude on difFeren 


t latitudes. 1 


Lat. 


Statute 


Naut. 


T "«- 


Statute 


Naut. 


T 




Statute 


Naut. 


Miles. 


Miles. 


J-. 


■ 


Miles. 


Miles. 


J^ctt. 


Miles. 


Miles. 


0^ 


' 


1.1527 


1.0000 


27° 


, 


1.0277 


.8916 


54° 


_, 


.6790 


.5891 




30 


1 . 1526 


.9999 




30 


1. 0232 


.8877 




30 


.6708 


.5820 


1 


— 


1.1525 


.9998 


28 


— 


1.0185 


.8836 


55 


— 


.6626 


.5749 




30 


1.1523 


.9996 




30 


1.0138 


.8795 




30 


.6544 


.5678 


2 


— 


1.1520 


.9994 


29 


— 


1.0089 


.8753 


56 


— 


.6460 


.5605 




30 


1.1516 


.9991 




30 


1.0041 


.8711 




30 


6377 


.5.533 


3 


— 


1.1511 


.9986 


30 


— 


.9991 


.8667 


57 


— 


.6293 


.54.59 




30 


1.1506 


.9981 




30 


.9940 


.8624 




30 


.6208 


.5416 


4 


— 


1.1499 


.9976 


31 


— 


.9889 


.8579 


58 


— 


.6123 


.5312 




30 


1.1491 


,9969 




30 


.9837 


.8534 




30 


.6048 


.5238 


5 


— 


1.1483 


.9962 


32 


— 


.9784 


.8488 


59 


— 


.5951 


.5161 




30 


1.1475 


.9954 




30 


.9731 


.8442 




30 


5865 


.5088 


6 


— 


1.1464 


.9945 


33 


— 


.9^76 


.8395 


60 


— 


.5778 


.5013 




30 


1.1453 


.9936 




30 


.9622 


.8348 




30 


.5691 


.4938 


7 


— 


1.1441 


.9926 


34 


— 


.9566 


.8299 


61 


— 


.5602 


.4861 




30 


1.1428 


.9915 




30 


.9509 


.8250 




30 


.5514 


.4785 


8 


— 


1.1415 


.9903 


35 


— 


.9452 


.8200 


62 


— 


.5425 


.4707 




30 


1.1401 


.9891 




30 


.9394 


.8150 




30 


.5337 


.4630 


9 


— 


1.1386 


.9877 


36 


— 


.9336 


.8099 


63 


— 


.5247 


.4552 




30 


1.1348 


.9863 




30 


.9277 


.8048 




30 


.5157 


.4474 


10 


— 


1.1352 


.9849 


37 


— 


.9217 


.7996 


64 


— 


5066 


.4395 




30 


1.1335 


.9833 




30 


.9157 


.7943 




30 


.4976 


.4317 


11 


— 


1.1316 


.9817 


38 


— 


.9095 


.7890 


65 


— 


.4885 


.4237 




30 


1.1297 


.9800 




30 


,9033 


.7837 




30 


.4794 


.4158 


12 


— 


1.1276 


.9783 


39 


— 


.8970 


.7782 


66 


— 


.4701 


.4079 




30 


1.1254 


.9764 




30 


.8907 


.7727 




30 


.4610 


.3999 


13 


— 


1.1233 


.9745 


40 




.8842 


.7671 


67 


— 


.4517 


.3918 




30 


1.1211 


.9726 




30 


.8778 


.7615 




30 


.4424 


.3838 


U 


— 


1.1187 


.9705 


41 





.8712 


.7558 


68 


— 


.4330 


.37.57 




30 


1.1163 


.9682 




30 


.8646 


.7501 




30 


.4238 


.3676 


15 


— 


1.1137 


.9661 


42 


— 


.8579 


.7443 


69 


— 


.4143 


.3594 




30 


1.1111 


.9638 




30 


.8512 


.7385 




30 


.4049 


.3513 


16 


— 


1.1083 


.9615 


43 




.8443 


,7325 


70 


— 


.3954 


.3430 




30 


1.1056 


.9571 




30 


.8375 


.7266 




30 


.3859 


.33.58 


17 


— 


1.1026 


.9566 


44 





.8305 


,7205 


71 


— 


.3764 


.3265 




30 


1.0997 


.9540 




30 


.8235 


.7143 




30 


.3668 


,3183 


18 


— 


1.0966 


.9513 


45 


— 


.8164 


.7080 


72 


— 


.3572 


.3099 




30 


1.0935 


.9486 




30 


.8093 


,7020 




30 


.3476 


.3016 


19 


— 


1.0903 


.9458 


46 




.8021 


,6960 


73 


— 


.3380 


.2933 




30 


1.0870 


.9430 




30 


.7949 


,6891 




30 


.3284 


.2850 


20 


— 


1.0836 


.9401 


47 


__ 


.7875 


.6831 


74 


— 


.3187 


.2765 




30 


1.0802 


.9372 




30 


.'J 802 


.6769 




30 


.3090 


.2681 


21 


— 


1.0766 


.9340 


48 


— 


.7727 


.6704 


75 


— 


.2993 


.2596 




30 


1 0728 


.9308 




30 


.7652 


.6639 




30 


.2895 


,2.512 


22 


— 


1.0692 


•9276 


40 




.7576 


.6573 


76 


— 


.2797 


.2427 




30 


1.0654 


.9243 




30 


.7500 


.6507 




30 


.2699 


.2343 


23 


— 


1.0616 


.9209 


50 


— 


.7422 


.6440 


77 


— 


.2601 


.2256 




30 


1.0577 


.9175 




30 


.7345 


.6347 




30 


.2503 


.2171 


24 


— 


1.0536 


.9140 


51 




.7268 


,6306 


78 


— 


.2404 


,2085 




30 


1.0495 


.9105 




30 


.7190 


.6238 




30 


,2305 


.2000 


25 


— 


1 0453 


.9069 


52 


— 


.7111 


,6169 


79 


— 


.2206 


.1914 




30 


1.0411 


.9032 




30 


.7032 


,6101 




30 


.2107 


.1823 


26 


— 


1.0367 


.8994 


53 




.6952 


.6031 


80 


— 


.2008 


,1742 





30 


1.0323 


.8056 




30 


.6872 


.5962 











153 



TABLE XI. 

This table contains the correction, in minutes, to be added 

to the middle latitude, to obtain the correct 

middle latitude. 


1^ 

^2 


DIFFERENCE OF LATITUDE. 


3° 


40 


5° 


50 70 


8° 


9° 


10° 


12° 


14° 


16° 


15^ 

18 

21 


2' 
1 

1 


3' 
3 

2 


5' 
4 
4 


7' 
6 
5 


9' 

8 

7 


12' 

10 

9 


15' 
13 
12 


18' 
16 
15 


26' 
23 
21 


36' 
32 
29 


47' 
41 

37 


24 
30 
35 


1 

1 
1 


2 

2 
2 


3 
3 
3 


5 
5 

4 


7 
6 
6 


9 

8 
8 


11 
10 
10 


14 
13 
12 


20 
18 

18 


27 
25 
24 


35 
32 
32 


40 
45 
50 


1 
1 
1 


2 
2 
2 


3 
3 
4 


5 
5 
5 


6 
6 

7 


8 
8 
9 


10 
10 
11 


13 
13 
14 


18 
18 
20 


25 

25 
28 


32 
32 
36 


55 

58 
60 


1 
2 
2 


3 
3 
3 


4 
4 
4 


6 
6 
6 


8 
8 
9 


10 
11 
11 


13 
14 
14 


16 
17 

18 


22 
24 
26 


31 
33 
35 


40 
43 
46 


62 
64 
66 


2 
2 
2 


3 
3 
4 


5 
5 
5 


7 
7 
8 


9 
10 
11 


12 
13 
14 


15 
16 
18 


19 
20 
22 


27 
29 
32 


37 
40 
43 


49 
52 
57 


This table is to be entered at the top with the 
Difference of the two latitudes, and at the side with the 
Middle Latitude; under the former, and opposite the 
latter, is the correction, in minutes, to be added to the 
middle latitude, to obtain the corrected middle latitude. 



154 



TABLE XII. 
Distance of Objects by Two Bearings. 

Useful in rounding headlands in the night. 



Ti tie 



P. 

3 

3% 

4 

4^ 

5 

5^ 

6 

6^ 

7 

75^ 



9 
10 



FIRST BEARINGS FROM HEADING OF 
VESSEL. — POINTS. 



2 2^ 3 3H 4 4M 5 5M 6 6K 7 7^ 8 8^ 



.5561. 

.6341 

.707 

.773 

.831 

.882 

.924 

.957 

.981 

.995 

.000 

.995 

.981 

.957 

.924 



322.42 
001.62 
8J 1.23 
691.00 
60 .85 



12.85 

1.91 

1.45 

1.18 

1.00 

.88 

.79 

.72 

.67 

.63 

.60 

.58 

.57 



25 

193.62 
66 2.44 3 

.35 1,85 3.66 
141.502.02 
001.27 1.64 
901.111.39 

.821.001.22 



.91 1.09 
.85 1.00 
.80| .93 
.77 .88 



I 

26 

2.86 4.52 
2.173.04 
1.76 2.30 
1.501.87 
1.311.59 
1.181.39 
1.08 1.25 
1.001.14 



4.74 

3.184.91 

2.413.30 5.03 

1.962.503.38 

1.662.032.57 

1.461. r2'2. 08 

1.311.511.77 



5.10 

3.43 5.13 

2.603.44J 

2.11 2. 6113. 435. ( 

I 



5.10 



Rui^E. — The number in column of first bearing, and 
in line of second bearing, multiplied by the distance seen 
between the times of taking the bearings, is the distance 
of object from the vessel at time of taking the second 
bearing, in units of the distance seen. 

And this distance multiplied by the sine of the 
second bearing, is the distance of the line of ship's course 
from the object, at right angles. 



155 





TABLE XIII. 


Table for Reducing Longitude to Time. 


o 


M. 




M. S. 


' 


M. S. 


/ 


M. S. 


' 


M. s. 


' 


M. S. 


' 


M. S. 


1 


4 


1 


4 


11 


.44 


21 


1.24 


31 


2.04 


41 


2.44 


51 


3.24 


2 


8 


2 


8 


12 


.48 


22 


1.28 


32 


2.08 


42 


2.48 


52 


3.28 


3 


12 


3 


12 


13 


.52 


23 


1.32 


33 


2.12 


43 


2.52 


53 


3.32 


4 


16 


4 


16 


14 


.56 


24 


1 36 


34 


2.16 


44 


2.56 


54 


3.36 


5 


20 


5 


20 


15 


1.00 


25 


1.40 


35 


2.20 


45 


3.00 


55 


3.40 


6 


24 


6 


24 


16 


1.04 


26 


1.44 


36 


2.24 


46 


3.04 


56 


3.44 


7 


28 


7 


28 


17 


1.08 


27 


1.48 


37 


2.28 


47 


3.08 


57 


3.48 


8 


32 


8 


32 


18 


1.12 


28 


1.52 


33 


2.32 


48 


3.12 


58 


3.52 


9 


36 


9 


36 


19 


1.16 


29 


1.56 


39 


2.36 


49 


3 16 


59 


3.56 


10 


40 


10 


40 


20 


1.20 


30 


2.00 


40 


2.40 


50 


3.20 


60 


4.00 




The above table is wanted for finding the error of 


the standard time watch, on local mean time, or on local 


apparent time, when expanding an amplitude into a 


tat 


>le of time azimuths of the sun, for the purpose of 


fin 


ding compass errors. 



156 



PLATE I.— PEARSON'S DIAGRAM. 



Var. East 



Vkr.We^t 




158 



PEARSON^S DIAGRAIYI.— PLATE I. 



Va.rBa.st Vkr.W^s^ 




159 



Se mA(UrtulcLr:an3L ' Qua^ranfal 

NortJ]Q ^ gee pa-ye ^ qSoujJi 




PLATE II. 



160 






^n 




__..^_... 



^ -* ^ > 



k^.-Zkt.^ 




161 



PLATE II. 



PLATE III. 




162 



2EARSON'S 
— DUMB 

COM R ASS. 




MANUFACTURED BY 

L. BEOKMANN, TOLEDO, O. 



Pearson's Dumb Compass 

MANUFACTURED AFTER THE /NVENTORS 
DIRECTIONS BY 

L. BEGKMANN, TOLEDO, 0. 



This little Instrument, so important to Masters and Pilots of 
Vessels, consists of a Tripod with Ball and Socket, on which 
turns a plate with index, which can be set parallel to the line of 
Keel. On this lower plate turns a circle divided into Degrees 
and Compass points, on which revolves an Alidade provided 
with an Index to read off the Degrees and Compass points; with 
a jDair of sight vanes and a pair of levels to set the whole instru- 
ment horizontal. 

Mostly made of brass and executed in the best manner and 
first-class workmanship. 

PRICE $30.00. 



FOR FURTHER PARTICULARS, ADDRESS 



L. BECKMANN, 

319 ADAMS ST., TOLEDO, O. 



Also keep in stock a full line of Aneroid Barometers, Marine 
Glasses and Compasses, Parallel Kulers, Etc. 



